A Flashback to the Morse Lemma ^†

Abstract

The Morse Lemma for a smooth function and the innovative version of the Morse Lemma for a holomorphic function are formulated and proved. Some new applications of the Morse Lemma are also discussed.

1. Introduction

The Morse Theory is the method of investigating functions on manifolds from the perspective of critical points of these functions. This theory was introduced by Marston Morse, and it has been the subject of interest of mathematicians, such as J. Milnor (e.g., [1]) and R. Palais and C. Terng (e.g., [2]). The Morse Theory is one of the most important parts of the critical real theory included in the Catastrophe Theory created by René Thom. Nowadays, the area of investigations in this field of mathematics is broad. For example, there are many scientific papers concerning Morse quasiflats—higher dimensional analogs of Morse quasigeodesics (e.g., [3])—or the behavior of finite Morse index solutions of some equations (e.g., [4]). Morse theory has also been considered for orbifolds or equivalently for differentiable Deligne–Mumford stacks. What is more, the Morse Lemma for differentiable Deligne–Mumford stacks was proved in [5] and describes the local form of a Morse function. Recently, in [6], a groupoid version of the Morse lemma has been shown to describe the topological behavior of the critical subgroupoid levels of a Morse Lie groupoid morphism around its non-degenerate critical orbits.

The aim of this paper is to show a very precise proof of the real and complex versions of the Morse Lemma and to give some new applications of these theorems. This paper is based on the author’s Master’s Dissertation ([7]) and was inspired by the work of J. Milnor: “Morse Theory. Based on lecture notes by M. Spivak and R. Wells” (see [1]). Included in the dissertation was a proof of the complex version of the Morse Lemma (Theorem 11), which in 2007 was original and new. Even today, it is difficult to find a fully detailed proof of the complex version of the Morse Lemma. The most valuable and innovative part of our proof of the complex version is the use of the branch of logarithm in the process of finding the wanted biholomorphism. The real version of the Morse Lemma (Theorem 10) is the fundamental fact of Morse Theory in a context of the notion of $C^{\infty }$-class functions index. Although the Morse Lemma is known in the literature (e.g., [8]), our presentation of proving the method—step by step—is clear and valuable. The technique for providing these considerations was partially presented by the author in 2021 at the conference (Hyper)Complex Seminar 2021 in memoriam of Professor Julian Ławrynowicz (see [9]). It is worth mentioning that this paper has been enriched with some new results, especially in Section 6.

Some necessary facts and definitions on linear algebra, complex analysis, and ordinary differential equations are included in Section 2.

Let f be a smooth function in a point $x^0\in {\mathbb{R}}^n$, i.e., f is a real function of $C^{\infty }$-class in an environment of $x^0\in {\mathbb{R}}^n$. We say that $x^0$ is a critical point of f, if the gradient $\mathrm{grad}f$ vanishes in $x^0$.

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a $C^{\infty }$-class function such that $f^{-1}\left([a,b]\right)$, where $a,b\in \mathbb{R}$, $a<b$, is a compact set not containing any critical points of f. Then, there exists a one-parameter group of diffeomorphisms $\phi :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, such that $\phi (\{b-a\}\times V_a)=V_b$, where $V_t=f^{-1}\left((-\infty ,t]\right)$ for $t\in \mathbb{R}$ (Theorem 9). Here, the one-parameter group of diffeomorphisms is constructed using the general solution of a corresponding system of ordinary differential equations. Under the assumptions of this theorem, the sets $f^{-1}\left(a\right)$ and $f^{-1}\left(b\right)$ are homeomorphic.

Among the critical points of a function f, we can distinguish the non-degenerate critical points, i.e., critical points $x^0\in {\mathbb{R}}^n$ of f, where

$$f\left(a\right)=0,\mathrm{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(x^0\right)\right]\ne 0.$$

In Section 4, we formulate and prove both versions of the Morse Lemma, as well as a few Lemmas (2)–(8) required to prove these two important theorems. First, we give the Morse Lemma for a smooth function.

Let f be a function of $C^{\infty }$-class in a certain environment of the point $a\in {\mathbb{R}}^n$, $n>1$, such that

$$f\left(a\right)=0,\mathrm{grad}f\left(a\right)=0,\mathrm{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ of $C^{\infty }$-class and the integer number $0⩽\mathit{l}⩽n$, such that

$$f\circ \phi \left(x\right)=-\sum _{i=1}^{\mathit{l}}{(x_i-a_i)}^2+\sum _{i=\mathit{l}+1}^n{(x_i-a_i)}^2$$

in a certain environment of the point a, where $a=(a_1,\dots ,a_n)$ and $x=(x_1,\dots ,x_n)$ are local coordinates centered at a.

The proof is the case when $x^0=0$. Then, we present a function f in the form $f\left(x\right)={\sum }_{i,j=1}^n{x}_i{x}_j{h}_{i,j}\left(x\right)$, where $h_{i,j}$ are $C^{\infty }$-class functions, and $det\left[h_{i,j}\left(0\right)\right]\ne 0$ (Lemma 4). Next, we use the method similar to the proof of Sylvester and Jacobi Theorem on the inertia of quadratic forms (see e.g., [10] (Theorem 5 §3.4 Section XI)). Here, Lemma 3 is an important element of this proof.

The non-degenerate critical points of a smooth function f are isolated (Corollary 3), i.e., for any non-degenerate critical point of f, there exists a neighborhood of this point in which f has no critical points (see also [6,7]). This result is an immediate consequence of the Morse Lemma.

Next, we similarly formulate and prove the Morse Lemma for a holomorphic function. Let f be a holomorphic function in a certain environment of the point $\stackrel{\mathrm{o}}{z}\in {\mathbb{C}}^n$, $n⩾1$, such that

$$f\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathrm{grad}f\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathrm{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{z}_l\mathsf{\partial }{z}_k}}\left(\stackrel{\mathrm{o}}{z}\right)\right]\ne 0.$$

Then, there exists a biholomorphism $\phi$ in a certain environment of the point $\stackrel{\mathrm{o}}{z}$, such that

$$f\circ \phi \left(z\right)=\sum _{k=1}^n{(z_k-{\stackrel{\mathrm{o}}{z}}_k)}^2\mathrm{in}\mathrm{a}\mathrm{certain}\mathrm{environment}\mathrm{of}\mathrm{the}\mathrm{point}\stackrel{\mathrm{o}}{z},$$

where $\stackrel{\mathrm{o}}{z}=({\stackrel{\mathrm{o}}{z}}_1,\dots ,{\stackrel{\mathrm{o}}{z}}_n)$.

Finally, we obtain Corollary 4 on the connection between meromorphic functions and the Laurent series and Corollary 5 for two holomorphic functions satisfying the Morse Lemma assumptions.

In Section 5, we introduce the notion of the index of a function f in a non-degenerate critical point $x^0$ as the index of bilinear form with a matrix $\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(x^0\right)\right]$. The index of f is the invariant of a diffeomorphism, i.e., it does not change for f composed with a diffeomorphism transforming $x^0$ to $x^0$ (Proposition 10).

In Section 6, we give some examples of exploitation of the Morse Lemma for some operators acting on symmetric forms in ${\mathbb{R}}^n$. Here, the influence of the Morse Lemma on the local shape of operators of a symmetric derivative (Corollary 6), gradient (Corollary 7), divergence (Corollary 8), $\mathrm{div}\mathrm{grad}$ (Proposition 18), and Sampson Laplacian (Proposition 19) is investigated. Additionally, we get two Corollaries for the $\mathrm{div}\mathrm{grad}$ operator and for the Sampson Laplacian (Corollary 9 and Corollary 10, respectively) on the influence of the Morse Lemma on solving some differential equations.

An interesting idea is the fact that, in the paper [11], Morse Theory is considered for perturbed Dirac-harmonic maps into flat tori. This research engages spin manifolds, so it provides the possibility for symmetric tensor considerations to evolve in some new fields of mathematics. What is more, the interest of physicists in models with various types of perturbations (see e.g., [11]) might provide some new applications for our considerations. Additionally, the Floer Theory (see e.g., [12]) is potentially an area of exploitation of the investigations in Section 6, especially in the context of symmetric tensors on symplectic manifolds. There are differential equations strictly related to the Morse Theory. The behavior of finite Morse index solutions of some equations can be studied in the context of stable solutions (see [4]). Another application is deriving such important properties of a solution like the number of critical points or the shape of the level sets (see [13]). Promising are also works on approximation theory that can build up the significance of our explorations in the future. Here, there are interesting directions of a study such as the development of wavelet-enhanced King-type Baskakov operators preserving quadratic test functions (see [14]), the convergence rate discussion using the Korovkin and Voronovskaja-type theorems, weighted approximation results, and statistical approximation theorems (see [15]), or Szász-Jakimovski–Leviatan Beta type operators, which are introduced through the Appell polynomials in Dunkl formulations (see [16]).

2. Preliminaries

Let $n\in \mathbb{N}$ and accept the convention that $0\notin \mathbb{N}$. For $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$, let $\parallel x\parallel =\sqrt{x_1^2+\cdots +x_n^2}$. We recall here some necessary facts.

2.1. Linear Spaces

Let $V,W$ be two vector spaces over $\mathbb{R}$. Let $F:V\times W\to \mathbb{R}$ be a bilinear mapping.

For $V=W={\mathbb{R}}^n$, let

$$F(v,w)=\sum _{i,j=1}^n{a}_{i,j}{v}_i{w}_j,$$

where $v=(v_1,\dots ,v_n),w=(w_1,\dots ,w_n)\in {\mathbb{R}}^n$,$a_{i,j}\in \mathbb{R}$, $i,j⩽n$. Then, F is called the bilinear form with the matrix $A={\left[a_{i,j}\right]}_{i,j=1,\dots ,n}$.

The bilinear form is called symmetric if it has a symmetric matrix.

Let $F:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ be a symmetric bilinear form, and let $V\subset {\mathbb{R}}^n$ be a linear subspace of the space ${\mathbb{R}}^n$. The form F is negatively (positively) defined on V, if $F(v,v)<0$ for all $v\in V\setminus \left\{0\right\}$ (respectively, $F(v,v)>0$ for all $v\in V\setminus \left\{0\right\}$).

If $V={\mathbb{R}}^n$ is negatively (positively) defined on V, form F is called negatively (positively) defined.

Let $G:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ be a symmetric bilinear form. The mapping $F:{\mathbb{R}}^n\to \mathbb{R}$, given by

$$F\left(v\right)=G(v,v)=\sum _{i,j=1}^n{a}_{i,j}{v}_i{v}_j,$$

where $v=(v_1,\dots ,v_n)\in {\mathbb{R}}^n$, $a_{i,j}\in \mathbb{R}$, $i,j⩽n$, is called the quadratic form.

Let V be a linear space, and let $V_1,\dots ,V_k$, $k\in \mathbb{N}$ be linear subspaces of V. As usual, we denote by $V_1+\dots +V_k$ and $V=V_1\oplus \dots \oplus V_k$ the sum and the direct sum of subspaces $V_1,\dots ,V_k$, respectively. (See e.g., [17]).

Theorem 1.

Let $V_1,V_2$ be two linear spaces over $\mathbb{R}$. Let $\phi :V_1\to V_2$ be a linear mapping. Then,

$$\dim V_1=\dim \left(\ker \phi \right)+\dim \left(\mathrm{im}\phi \right),$$

where $\ker \phi =\{\alpha \in V_1:\phi \left(\alpha \right)=0\}$, and $\mathrm{im}\phi =\{\beta \in V_2:\beta =\phi \left(\alpha \right),\alpha \in V_1\}$.

Proof. 

See, e.g., [17]. □

Proposition 1.

Let $V_1$ and $V_2$ be two linear subspaces of finite dimensional vector space V. Then,

$$\dim (V_1\cap V_2)⩾\dim V_1+\dim V_2-\dim V.$$

Proof. 

See, e.g., [17]. □

2.2. Complex Analysis

For $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$, let

$$\sum _{n=-\infty }^{+\infty }{a}_n{(z-\stackrel{\mathrm{o}}{z})}^n$$

(1)

be the so-called Laurent series, where

$$\sum _{n=0}^{\infty }{a}_n{(z-\stackrel{\mathrm{o}}{z})}^n$$

(2)

and

$$\sum _{n=1}^{\infty }{a}_{-n}/{(z-\stackrel{\mathrm{o}}{z})}^n$$

(3)

are the regular and the main part of the Laurent series, respectively.

Let f be a holomorphic function in a neighborhood of a point $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$. Then, there is a neighborhood U of a point $\stackrel{\mathrm{o}}{z}$, such that

$$f\left(z\right)=\sum _{n=-\infty }^{+\infty }{a}_n{(z-\stackrel{\mathrm{o}}{z})}^n,z\in U$$

synonymous (see [18]).

Notice that, as an environment of the point $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$, we understand any set $\{z\in \mathbb{C}:|z-\stackrel{\mathrm{o}}{z}|<r\}$, $r>0$, and as a neighborhood of the point $\stackrel{\mathrm{o}}{z}$, we understand any set $\{z\in \mathbb{C}:0<|z-\stackrel{\mathrm{o}}{z}|<r\}$, $r>0$ (see [18]). This notation will be retained.

The point $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$ is called an apparent critical point of f, if the main part (3) of the Laurent series (1) vanishes.

The point $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$ is called a k-pole of f, if the main part (3) of the Laurent series (1) is of the form

$$\sum _{n=1}^k{a}_{-n}/{(z-\stackrel{\mathrm{o}}{z})}^n,\mathrm{where}{a}_{-k}\ne 0.$$

A function f is called meromorphic in a point $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$, if f is holomorphic in some neighborhood of $\stackrel{\mathrm{o}}{z}$, and if $\stackrel{\mathrm{o}}{z}$ is an apparent critical point or a pole of f.

Let f be a meromorphic function in $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$ not vanishing identically in any neighborhood of $\stackrel{\mathrm{o}}{z}$. Then, a function f, in some neighborhood of $\stackrel{\mathrm{o}}{z}$, is of the form

$$f\left(z\right)=\sum _{n=l}^{\infty }{a}_n{(z-\stackrel{\mathrm{o}}{z})}^n,\mathrm{where}{a}_l\ne 0\mathrm{for}\mathrm{some}l\in \mathbb{Z}.$$

The integer l is called an order of f in $\stackrel{\mathrm{o}}{z}$ and denoted by ${\mathrm{ord}}_{\stackrel{\mathrm{o}}{z}}f$ (see [18]).

Corollary 1.

Let f be a meromorphic function in $\stackrel{\mathrm{o}}{z}$ not vanishing identically in any neighborhood of $\stackrel{\mathrm{o}}{z}$. Then, $1/f$ is also a meromorphic function in $\stackrel{\mathrm{o}}{z}$.

Proof. 

See, e.g., [18]. □

Let f be a holomorphic function in $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$ not vanishing identically in any neighborhood of $\stackrel{\mathrm{o}}{z}$. If $f\left(\stackrel{\mathrm{o}}{z}\right)=0$, then ${\mathrm{ord}}_{\stackrel{\mathrm{o}}{z}}f=k>0$, and $\stackrel{\mathrm{o}}{z}$ is called a k-root of f (see [18]).

Corollary 2.

A point $\stackrel{\mathrm{o}}{z}$ is a k-root of a function f if and only if $\stackrel{\mathrm{o}}{z}$ is a k-pole of a function $1/f$.

Proof. 

See, e.g., [18]. □

Let $\stackrel{\mathrm{o}}{z}\in {\mathbb{C}}^n$. A function f defined in a certain environment of the point $\stackrel{\mathrm{o}}{z}$ is $\mathbb{C}$-differentiable in $\stackrel{\mathrm{o}}{z}$, if there exists an environment U of the point $\stackrel{\mathrm{o}}{z}$ and some functions $f_j:U\to \mathbb{C}$, $j=1,\dots ,n$, continuous in $\stackrel{\mathrm{o}}{z}$, such that

$$f\left(z\right)=f\left(\stackrel{\mathrm{o}}{z}\right)+\sum _{j=1}^n(z_j-{\stackrel{\mathrm{o}}{z}}_j)f_j\left(z\right)\mathrm{for}z\in U,$$

(4)

where $z=(z_1,\dots ,z_n)$, $\stackrel{\mathrm{o}}{z}=({\stackrel{\mathrm{o}}{z}}_1,\dots ,{\stackrel{\mathrm{o}}{z}}_n)$.

A function f is $\mathbb{C}$-differentiable in a set $G\subset {\mathbb{C}}^n$, if f is $\mathbb{C}$-differentiable in any point of G.

Proposition 2.

Let f be a $\mathbb{C}$-differentiable function in $\stackrel{\mathrm{o}}{z}\in {\mathbb{C}}^n$. Then, f is continuous in $\stackrel{\mathrm{o}}{z}$, and there exist ${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{z}_j}}\left(\stackrel{\mathrm{o}}{z}\right)$ for all $j=1,\dots ,n$. What is more,

$${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{z}_j}}\left(\stackrel{\mathrm{o}}{z}\right)=f_j\left(\stackrel{\mathrm{o}}{z}\right)\mathit{for}j=1,\dots ,n,$$

where $f_j$ is such as in (4).

Proof. 

See, e.g., [19]. □

Let $z=(z_1,\dots ,z_n),\stackrel{\mathrm{o}}{z}=({\stackrel{\mathrm{o}}{z}}_1,\dots ,{\stackrel{\mathrm{o}}{z}}_n)\in {\mathbb{C}}^n$ and $r_j\in \mathbb{R}$, $r_j>0$ for $j=1,\dots ,n$. We denote by P the policylinder $\{z\in {\mathbb{C}}^n:|z_j-{\stackrel{\mathrm{o}}{z}}_j|<r_j,j=1,\dots ,n\}$ with center at $\stackrel{\mathrm{o}}{z}$ and radiuses $r_1,\dots ,r_n$.

Lemma 1.

Let f be a holomorphic function in a policylinder P. Then, there exist holomorphic functions $f_j:P\to \mathbb{C}$, $j=1,\dots ,n$, such that

$$f\left(z\right)=f\left(\stackrel{\mathrm{o}}{z}\right)+\sum _{j=1}^n(z_j-{\stackrel{\mathrm{o}}{z}}_j)f_j\left(z\right)\mathit{for}z\in P.$$

Proof. 

See, e.g., [19]. □

Theorem 2.

Let f be a function defined in a certain environment of $\stackrel{\mathrm{o}}{z}\in {\mathbb{C}}^n$. The following conditions are equivalent:

(a)

f is holomorphic in $\stackrel{\mathrm{o}}{z}$,

(b)

f is $\mathbb{C}$—differentiable in some environment of $\stackrel{\mathrm{o}}{z}$,

(c)

f has a power series (with center at $\stackrel{\mathrm{o}}{z}$) representation in some environment of $\stackrel{\mathrm{o}}{z}$.

Proof. 

See, e.g., [19]. □

Let $G,D\subset {\mathbb{C}}^n$ be open sets. Recall that a mapping $f:G\to D$ is called the biholomorphism, if f is a holomorphic bijection. Then, $f^{-1}$ is also a holomorphic mapping. What is more, the inverse mapping and a composition of biholomorphisms are biholomorphic mappings (see [19]).

Theorem 3.

Let $G\subset {\mathbb{C}}^n$ be an open set, $f:G\to {\mathbb{C}}^n$ be a holomorphic mapping, and

$$det{\left[(\mathsf{\partial }{f}_k/\mathsf{\partial }{z}_j)\left(\stackrel{\mathrm{o}}{z}\right)\right]}_{k,j=1,\dots ,n}\ne 0\mathit{for}\mathit{certain}\stackrel{\mathrm{o}}{z}\in G.$$

Then, there exist an environment U of $\stackrel{\mathrm{o}}{z}$ and an environment V of the point $f\left(\stackrel{\mathrm{o}}{z}\right)$, such that $f_{|U}:U\to V$ is one-to-one mapping, and the mapping ${\left(f_{|U}\right)}^{-1}:V\to U$ is holomorphic.

Proof. 

See, e.g., [19]. □

Let $P\subset {\mathbb{C}}^n$ be a policylinder and $f:P\to \mathbb{C}\setminus \left\{0\right\}$ be a continuous function.

Theorem 4.

There exists a continuous function $\tilde{f}:P\to \mathbb{C}$, such that $exp\circ \tilde{f}=f$, i.e., there exists a branch of logarithm of f in P. Additionally, if f is holomorphic, then $\tilde{f}$ is also holomorphic.

Proof. 

See, e.g., [19]. □

2.3. Some Facts About Ordinary Differential Equations

Let $G\subset \mathbb{R}\times {\mathbb{R}}^n$ be an open set.

Consider the following system of differential equations

$$y^{\prime }=F(t,y),$$

(5)

where the right hand side is defined in G, and $y=(y_1,\dots ,y_n)$, $y^{\prime }=(y_1^{\prime },\dots ,y_n^{\prime })$.

A solution of system (5) is any differentiable mapping $\phi :I\to {\mathbb{R}}^n$, where $I\subset \mathbb{R}$ is an interval, such that

$$(t,\phi \left(t\right))\in G\mathrm{and}{\phi }^{\prime }\left(t\right)=F(t,\phi \left(t\right))\mathrm{for}\mathrm{all}t\in I.$$

Theorem 5.

Let $F:G\to {\mathbb{R}}^n$ be a continuous mapping. Then, for any point $(\tau ,\eta )\in G$, there exists a solution $\phi :I\to {\mathbb{R}}^n$ of system (5), such that $\tau \in I$ and $\phi \left(\tau \right)=\eta$.

Proof. 

See, e.g., [20,21]. □

Let $(\tau ,\eta )\in G$ be a fixed point. Recall that the problem of finding a solution $\phi :I\to {\mathbb{R}}^n$ to system (5), such that $\tau \in I$ and

$$\phi \left(\tau \right)=\eta ,$$

(6)

is called the Cauchy problem.

Let $\phi :I\to {\mathbb{R}}^n$ be a solution to system (5). Then, the solution $\tilde{\phi }:\tilde{I}\to {\mathbb{R}}^n$ is an extension of $\phi$, when

$$I\subset \tilde{I}and{\tilde{\phi }}_{|I}=\phi .$$

If $I\ne \tilde{I}$, then $\tilde{\phi }$ is called the proper extension.

A solution to system (5), which has no proper extensions is called the integral solution to system (5).

Proposition 3.

Let $F:G\to {\mathbb{R}}^n$ be a continuous mapping. Then, any integral solution to system (5) is defined on an open interval.

Proof. 

See, e.g., [20,21]. □

Theorem 6.

Let $F:G\to {\mathbb{R}}^n$ be a continuous mapping. Let $\phi :(a,b)\to {\mathbb{R}}^n$ be the integral solution to system (5). Then, for any compact set $K\subset G$, there exists a closed interval $I\subset (a,b)$, such that

$$(t,\phi (t\left)\right)\in G\setminus Kforallt\in (a,b)\setminus I.$$

Proof. 

See, e.g., [22]. □

For the system (5) holds the global uniqueness of solutions, if for any point $(\tau ,\eta )\in G$, there exists only one integral solution $\phi (\tau ,\eta ):I(\tau ,\eta )\to {\mathbb{R}}^n$ to system (5). Here, $I(\tau ,\eta )\subset \mathbb{R}$ is an open interval, such that the initial terms $\phi (\tau ,\eta )\left(\tau \right)=\eta$ hold.

Assume that for system (5) the global uniqueness of solutions holds. Let

$$V=\{(\tau ,\eta ,t)\in \mathbb{R}\times {\mathbb{R}}^n\times \mathbb{R}:(\tau ,\eta )\in G,t\in I(\tau ,\eta )\},$$

and let $\varphi :V\to {\mathbb{R}}^n$ be the mapping given by

$$\varphi (\tau ,\eta ,t)=\phi (\tau ,\eta )\left(t\right),(\tau ,\eta ,t)\in V.$$

A mapping $\varphi$ is called the general solution to system (5).

Proposition 4.

Let $F:G\to {\mathbb{R}}^n$ be a mapping of $C^1$-class. Then, for system (5), the global uniqueness of solutions holds, the general solution is well defined, and its domain is an open set.

Proof. 

See, e.g., [20,21]. □

Proposition 5.

Let ϕ be the general solution to system (5). Then, for any $({\tau }_1,{\eta }_1),({\tau }_2,{\eta }_2)\in G$, we get

$$\varphi ({\tau }_1,{\eta }_1,{\tau }_1)={\eta }_1,I({\tau }_1,{\eta }_1)=I({\tau }_2,{\eta }_2),$$

$$\varphi ({\tau }_1,{\eta }_1,t)=\varphi ({\tau }_2,{\eta }_2,t)fort\in I({\tau }_1,{\eta }_1),$$

where ${\tau }_2\in I({\tau }_1,{\eta }_1)$, ${\eta }_2=\varphi ({\tau }_1,{\eta }_1,{\tau }_2)$.

Proof. 

See, e.g., [20,21]. □

Theorem 7.

Let $m\in \mathbb{N}$, and let $F:G\to {\mathbb{R}}^n$ be a $C^m$-class mapping. Then, the general solution $\varphi :V\to {\mathbb{R}}^n$ to system (5) is a $C^m$-class mapping.

Proof. 

See, e.g., [21,23,24]. □

Now, we prove the following:

Proposition 6.

Let $X:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a vector field of $C^{\infty }$-class, and let $\varphi :\mathbb{R}\times {\mathbb{R}}^n\times \mathbb{R}\to {\mathbb{R}}^n$ be the general solution to the system:

$$\dot{x}=X\left(x\right).$$

(7)

Then, for any $b\in \mathbb{R}$ and $\eta \in {\mathbb{R}}^n$, the following holds:

$$\varphi (0,\eta ,t)=\varphi (b,\eta ,t+b),t\in \mathbb{R}.$$

Proof. 

Let $\phi (0,\eta ):\mathbb{R}\to {\mathbb{R}}^n$ be the integral solution to system (7), such that

$$\phi (0,\eta )\left(0\right)=\eta .$$

Let $\phi (b,\eta ):\mathbb{R}\to {\mathbb{R}}^n$ be the integral solution to system (7), such that

$$\phi (b,\eta )\left(b\right)=\eta .$$

Define the mapping $\psi :\mathbb{R}\to {\mathbb{R}}^n$ by

$$\psi \left(t\right)=\phi (b,\eta )(t+b),t\in \mathbb{R}.$$

Then,

$${\psi }^{\prime }\left(t\right)={\left(\phi (b,\eta )\right)}^{\prime }(t+b)\mathrm{and}\psi \left(0\right)=\phi (b,\eta )\left(b\right)=\eta .$$

Let us now prove that

$${\psi }^{\prime }\left(t\right)=X\left(\psi \left(t\right)\right).$$

We have

$${\left(\phi (b,\eta )\right)}^{\prime }\left(\xi \right)=X\left(\phi (b,\eta )\left(\xi \right)\right)\xi \in \mathbb{R}.$$

Let $\xi =t+b$, $t\in \mathbb{R}$. Then,

$${\psi }^{\prime }\left(t\right)={\left(\phi (b,\eta )(t+b)\right)}^{\prime }={\left(\phi (b,\eta )\right)}^{\prime }(t+b)=X\left(\phi (b,\eta )(t+b)\right)=X\left(\psi \left(t\right)\right).$$

Therefore, $\psi$ is the integral solution to system (7), such that $\psi \left(0\right)=\eta$. So, by the global uniqueness of solutions, we get

$$\phi (0,\eta )\left(t\right)=\psi \left(t\right)=\phi (b,\eta )(t+b)\mathrm{for}t\in \mathbb{R},$$

which ends the proof. □

3. The One-Parameter Group of Diffeomorphisms

First, recall that the one-parameter group of diffeomorphisms of the space ${\mathbb{R}}^n$ is the mapping $\phi :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ of $C^{\infty }$- class, such that

(a)

For any $t\in \mathbb{R}$, the mapping ${\phi }_t:{\mathbb{R}}^n\to {\mathbb{R}}^n$, given by

$${\phi }_t\left(x\right)=\phi (t,x),x\in {\mathbb{R}}^n,$$

is a diffeomorphism, i.e., mappings ${\phi }_t$ and ${\phi }_t^{-1}$ are of $C^{\infty }$-class,

(b)

For all $t,s\in \mathbb{R}$, we have

$${\phi }_{t+s}={\phi }_t\circ {\phi }_s.$$

Proposition 7.

Let $\phi :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ be the one-parameter group of diffeomorphisms. Then, the mapping ${\phi }_0:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is the identity.

Proof. 

For any $x^0\in {\mathbb{R}}^n$, using $\left(b\right)$ in definition of the one-parameter group of diffeomorphisms, we have

$${\phi }_0\left(x^0\right)={\phi }_{0+0}\left(x^0\right)={\phi }_0\left({\phi }_0\left(x^0\right)\right).$$

So, by the injectivity of ${\phi }_0$, we get ${\phi }_0\left(x^0\right)=x^0.$ □

Let $\phi$ be the one-parameter group of diffeomorphisms of ${\mathbb{R}}^n$. A vector field X on ${\mathbb{R}}^n$ is generating the group $\phi$, if

$${\displaystyle \frac{d}{dt}}\phi (t,x)=X\left(\phi (t,x)\right),\mathrm{for}t\in \mathbb{R},x\in {\mathbb{R}}^n.$$

Theorem 8.

Let $X:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a vector field of $C^{\infty }$- class, such that X vanishes outside a certain compact set. Let $\varphi :V\to {\mathbb{R}}^n$ be the general solution to the system

$$\dot{x}=X\left(x\right),$$

(8)

where $V=\{(\xi ,\eta ,t)\in \mathbb{R}\times {\mathbb{R}}^n\times \mathbb{R},t\in I(\xi ,\eta )\}$. Then,

$$V=\mathbb{R}\times {\mathbb{R}}^n\times \mathbb{R}$$

and the mapping $\phi :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ given by

$$\phi (a,\eta )=\varphi (0,\eta ,a)$$

is one-parameter group of diffeomorphisms of ${\mathbb{R}}^n$.

Proof. 

Let $\psi :I(\xi ,\eta )\to {\mathbb{R}}^n$ be the integral solution to system (8), such that $\psi \left(\xi \right)=\eta$. By Proposition 3, $I(\xi ,\eta )$ is an open interval. Let $I(\xi ,\eta )=(\alpha ,\beta )$. We prove that $\beta =+\infty$. Suppose conversely that $\beta <+\infty$. Let $p\in (\alpha ,\beta )$, and let $K\subset {\mathbb{R}}^n$ be a compact set, such that

$$X\left(x\right)=0\mathrm{for}x\in {\mathbb{R}}^n\setminus K.$$

By Theorem 6, there exists an interval $[r,s]\subset (\alpha ,\beta )$, such that

$$(t,\psi \left(t\right))\in \mathbb{R}\times {\mathbb{R}}^n\setminus ([p,\beta ]\times K)\mathrm{for}t\in (\alpha ,\beta )\setminus [r,s].$$

Then, for $t\in \left[\text{max}\right\{s,p\},\beta )$, we have $\psi \left(t\right)\notin K$, so ${\psi }^{\prime }\left(t\right)=0$. Therefore, $\psi$ is a constant function in the interval $\left[\text{max}\right\{s,p\},\beta )$. By the uniqueness of solutions to system (8), we get that $\psi$ is a constant mapping, and $(\alpha ,\beta )=\mathbb{R}$. This is in contradiction to the assumption that $\beta <+\infty$. So, $\beta =+\infty$. We prove that $\alpha =-\infty$ by analogy. Finally, $V=\mathbb{R}\times {\mathbb{R}}^n\times \mathbb{R}$.

Let now prove the second part of the assertion. By Proposition 4, for system (8), the global uniqueness of the solutions holds. So, the mapping $\phi$ is well-defined. By Theorem 7, the general solution $\varphi$ is of $C^{\infty }$-class; so, the mapping $\phi$ is also of $C^{\infty }$-class. What is more, $\phi$ is the one-parameter group of diffeomorphisms of ${\mathbb{R}}^n$. Indeed, for any $a,b\in \mathbb{R}$ and $\eta \in {\mathbb{R}}^n$, by Propositions 5, 6, for ${\tau }_1=0$, ${\tau }_2=b$, we get

$$\begin{array}{cc}\hfill {\phi }_a\circ {\phi }_b\left(\eta \right)& =\varphi (0,{\phi }_b\left(\eta \right),a)=\varphi (0,\varphi (0,\eta ,b),a)=\varphi (b,\varphi (0,\eta ,b),a+b)\hfill \\ \hfill & =\varphi (0,\eta ,a+b)={\phi }_{a+b}\left(\eta \right).\hfill \end{array}$$

Additionally, we have

$${\phi }_0\left(\eta \right)=\varphi (0,\eta ,0)=\eta ,\mathrm{for}\mathrm{any}\eta \in {\mathbb{R}}^n.$$

Therefore, ${\phi }_a$ for $a\in \mathbb{R}$ is a diffeomorphism, where ${\phi }_a^{-1}={\phi }_{-a}$. So, $\phi$ is one-parameter group of diffeomorphisms of ${\mathbb{R}}^n$. □

Remark 1.

A one-parameter group of diffeomorphisms φ defined in Theorem 8 is the one-parameter group of diffeomorphisms assigned by the vector field X.

Let f be a real function of $C^{\infty }$-class in an open set $G\subset {\mathbb{R}}^n$. A point $x^0\in G$ is called the critical point of f, if

$$\mathrm{grad}f\left(x^0\right)=\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{x}_1}}\left(x^0\right),\dots ,{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{x}_n}}\left(x^0\right)\right)=0.$$

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ and $a\in \mathbb{R}$. We denote

$$V_a=\{x\in {\mathbb{R}}^n:f\left(x\right)⩽a\}=f^{-1}\left((-\infty ,a]\right).$$

Theorem 9.

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a $C^{\infty }$-class function and let $a,b\in \mathbb{R}$, $a<b$. If $f^{-1}\left([a,b]\right)$ is a compact set having no critical points, then there exists a one-parameter group of diffeomorphisms $\phi :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, such that

$${\phi }_{b-a}\left(V_a\right)=V_b.$$

(9)

Proof. 

The set $f^{-1}\left([a,b]\right)$ has no critical points, and the set $\{x\in {\mathbb{R}}^n:\mathrm{grad}f\left(x\right)=0\}$ is closed; so, there exists an open set $U\subset {\mathbb{R}}^n$, such that

$$f^{-1}\left([a,b]\right)\subset Uand\mathrm{grad}f\left(x\right)\ne 0forx\in U.$$

By the fact that the set $f^{-1}\left([a,b]\right)$ is compact, we can assume that the set U is bounded and that there exists an open set $W\subset {\mathbb{R}}^n$, such that

$$f^{-1}\left([a,b]\right)\subset Wand\overline{W}\subset U.$$

Then, there exists a function $g:{\mathbb{R}}^n\to \mathbb{R}$ of $C^{\infty }$- class, such that

$$g\left(x\right)=\left\{\begin{array}{c}1\mathrm{for}x\in f^{-1}\left([a,b]\right),\hfill \\ 0\mathrm{for}x\in {\mathbb{R}}^n\setminus W.\hfill \end{array}\right.$$

Let $X:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a vector field given by

$$X\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{grad}f\left(x\right)}{{\parallel \mathrm{grad}f\left(x\right)\parallel }^2}}g\left(x\right)\mathrm{for}x\in W,\hfill \\ 0\mathrm{for}x\in {\mathbb{R}}^n\setminus W.\hfill \end{array}\right.$$

Then, X is of $C^{\infty }$- class and vanishes outside a compact set $\overline{W}$. Consider now the system of equations

$$\dot{x}=X\left(x\right).$$

(10)

Let $\varphi$ be the general solution to system (10). By Theorem 8, any integral solution of the system (10) is defined in $\mathbb{R}$. Moreover, for $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$, such that $\varphi (0,x,t)\in f^{-1}\left([a,b]\right)$, we have

$${\displaystyle \frac{d}{dt}}f\circ \varphi (0,x,t)=1.$$

(11)

Indeed, $\gamma (0,x)\left(t\right)=\varphi (0,x,t)$, $t\in \mathbb{R}$ is the integral solution to system (10); so,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}f\circ \varphi (0,x,t)& =〈\mathrm{grad}f\left(\gamma (0,x)\left(t\right)\right),{\gamma }^{\prime }(0,x)\left(t\right)〉\hfill \\ \hfill & =〈\mathrm{grad}f\left(\gamma \right(0,x\left)\right(t\left)\right),X\left(\gamma \right(0,x\left)\right(t\left)\right)〉\hfill \\ \hfill & =〈\mathrm{grad}f\left(\gamma (0,x)\left(t\right)\right),{\displaystyle \frac{\mathrm{grad}f\left(\gamma \right(0,x\left)\right(t\left)\right)}{{\parallel \mathrm{grad}f\left(\gamma (0,x)\left(t\right)\right)\parallel }^2}}〉=1.\hfill \end{array}$$

Therefore, we get (11).

Let $\phi$ be a one-parameter group of diffeomorphisms assigned by the field X. Consider the diffeomorphism ${\phi }_{b-a}:{\mathbb{R}}^n\to {\mathbb{R}}^n$.

First, we prove that

$${\phi }_{b-a}\left(V_a\right)\subset V_b.$$

(12)

For any $y\in {\phi }_{b-a}\left(V_a\right)$, there exists $x\in V_a$, such that $y={\phi }_{b-a}\left(x\right)$. Let $\gamma (0,x):\mathbb{R}\to {\mathbb{R}}^n$ be the integral solution of the system (10), such that $\gamma (0,x)\left(0\right)=x$. Define the function $\psi :\mathbb{R}\to \mathbb{R}$ by

$$\psi \left(t\right)=f\circ \gamma (0,x)\left(t\right)fort\in \mathbb{R}.$$

If $x\notin W$, then the solution $\gamma (0,x)$ is constant. So,

$$y={\phi }_{b-a}\left(x\right)=\gamma (0,x)(b-a)=\gamma (0,x)\left(0\right)=x\in V_a\subset V_b.$$

Let $x\in W$. If $\psi \left(t\right)<a$ for $t\in [0,b-a]$, then

$$y={\phi }_{b-a}\left(x\right)=\gamma (0,x)(b-a)\in V_a\subset V_b.$$

Suppose that there exists $t_0\in [0,b-a]$, such that $\psi \left(t_0\right)⩾a$. Then, by the fact that $x\in V_a$, there exists $t_1\in [0,b-a]$, such that $\psi \left(t_1\right)=a$. Let

$$t_2=min\{t\in [0,b-a]:\psi \left(t\right)=a\}.$$

Suppose now that $\gamma (0,x)(b-a)={\phi }_{b-a}\left(x\right)\notin V_b$, i.e.,

$$\psi (b-a)>b.$$

(13)

Let

$$t_3=inf\{t\in [0,b-a]:\psi \left(t\right)>b\}.$$

Then, by the fact that $\psi \left(t_2\right)=a$ and $\psi \left(t_3\right)=b$, using (11), we get $t_2<t_3$ and

$$\gamma (0,x)\left(t\right)\in f^{-1}\left([a,b]\right)fort\in [t_2,t_3].$$

Additionally, $\psi \left(t\right)=t+B$ for $t\in [t_2,t_3]$, where $B\in \mathbb{R}$ and $B=\psi \left(t_2\right)-t_2$. So,

$$\psi \left(t\right)=\psi \left(t_2\right)+t-t_2=a+t-t_2\mathrm{for}t\in [t_2,t_3].$$

By (13) and by the definition of $t_3$, we get that $t_3<b-a$. By the fact that $t_2⩾0$, we get

$$\psi \left(t_3\right)<a+b-a-t_2=b-t_2⩽b,$$

which is in contradiction to the definition of $t_3$ and gives

$$y={\phi }_{b-a}\left(x\right)\in V_b.$$

By the arbitrary of choice of $y\in {\phi }_{b-a}\left(V_a\right)$, we get (12).

Now, let us prove that

$$V_b\subset {\phi }_{b-a}\left(V_a\right).$$

(14)

For any $y\in V_b$, we get $f\left(y\right)⩽b$. First, we prove that there exists $x\in V_a$, such that $y={\phi }_{b-a}\left(x\right)$. It is enough to show that $x={\phi }_{a-b}\left(y\right)\in V_a$. Let $\gamma (0,y):\mathbb{R}\to {\mathbb{R}}^n$ be the integral solution to system (10), such that $\gamma (0,y)\left(0\right)=y$. Define the function $\overline{\psi }:\mathbb{R}\to \mathbb{R}$ by

$$\overline{\psi }\left(t\right)=f\circ \gamma (0,y)\left(t\right)fort\in \mathbb{R}.$$

If $y\notin W$, then $f\left(y\right)⩽b$ and $y\notin f^{-1}\left([a,b]\right)$; then, $f\left(y\right)<a$, and $\gamma (0,y)\left(t\right)=y$, $t\in \mathbb{R}$ is the integral solution to system (10). Therefore,

$$x={\phi }_{a-b}\left(y\right)=\gamma (0,y)(a-b)=y\in V_a.$$

Let $y\in W$. If $\overline{\psi }\left(t\right)\in [a,b]$ for $t\in [a-b,0]$, then by (11),

$${\overline{\psi }}^{\prime }\left(t\right)=1\mathrm{for}t\in [a-b,0].$$

So,

$$\overline{\psi }\left(t\right)=f\left(y\right)+t⩽b+t\mathrm{for}t\in [a-b,0].$$

Therefore, for $t=a-b$, we have $\overline{\psi }(a-b)⩽b+a-b=a$, and

$$x={\phi }_{a-b}\left(y\right)=\gamma (0,y)(a-b)\in V_a.$$

Suppose that there exists $s_0\in [a-b,0]$, such that $\overline{\psi }\left(s_0\right)<a$. Let us now prove that $\overline{\psi }(a-b)⩽a$. Suppose, conversely, that $\overline{\psi }(a-b)>a$. Let

$$s_1=min\{t\in [a-b,0]:\overline{\psi }\left(t\right)=a\}.$$

Then, ${\overline{\psi }}^{\prime }\left(s_1\right)=1$; so,

$$\overline{\psi }\left(t\right)<\overline{\psi }\left(s_1\right)=a,\mathrm{for}t\in [a-b,s_1].$$

This is in contradiction to the assumption that $\overline{\psi }(a-b)>a$. So, taking $x={\phi }_{a-b}\left(y\right)$, we get

$$f\left(x\right)=f\left({\phi }_{a-b}\left(y\right)\right)=f\left(\gamma (0,y)(a-b)\right)=\overline{\psi }(a-b)⩽a.$$

Therefore, $x={\phi }_{a-b}\left(y\right)\in V_a$. Suppose now that there exists $s\in [a-b,0]$, such that $\overline{\psi }\left(s\right)>b$. We show that this case does not hold. Let

$$s_2=sup\{t\in [a-b,0]:\overline{\psi }\left(t\right)>b\}.$$

Then, we have

$$\overline{\psi }\left(t\right)⩽b\mathrm{for}t\in [s_2,0]\mathrm{and}\overline{\psi }\left(s_2\right)=b.$$

So, by (11), ${\overline{\psi }}^{\prime }\left(s_2\right)=1$. Therefore, there exists $\delta >0$, such that

$$\overline{\psi }\left(t\right)<\overline{\psi }\left(s_2\right)=b\mathrm{for}t\in (s_2-\delta ,s_2),$$

which gives that

$$\overline{\psi }\left(t\right)⩽b\mathrm{for}t\in (s_2-\delta ,0],$$

which is in contradiction to the definition of $s_2$; so, this case does not hold. We proved that ${\phi }_{a-b}\left(y\right)\in V_a$; so, (14) holds.

Finally, by (12) and (14), we have the assertion. □

4. Morse Lemma

4.1. Morse Lemma for a Smooth Function

Let $a=(a_1,\dots ,a_n)\in {\mathbb{R}}^n$. We denote by $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^s,b)$ a mapping $\phi$ defined in a certain environment of the point $a\in {\mathbb{R}}^n$ with values in ${\mathbb{R}}^s$ such that $\phi \left(a\right)=b$.

Theorem 10. Morse Lemma.

Let f be a smooth function (of $C^{\infty }$-class) in a certain environment of the point $a\in {\mathbb{R}}^n$, $n>1$, such that

$$f\left(a\right)=0,\mathrm{grad}f\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ of $C^{\infty }$-class and the integer number $0⩽\mathit{l}⩽n$, such that

$$f\circ \phi \left(x\right)=-\sum _{i=1}^{\mathit{l}}{(x_i-a_i)}^2+\sum _{i=\mathit{l}+1}^n{(x_i-a_i)}^2$$

in a certain environment of a.

First, we are going to prove the following lemmas.

Lemma 2.

Let $D\subset {\mathbb{R}}^n$ be an open set and $\phi :[0,1]\times D\to \mathbb{R}$ be a continuous function. Then, the function $\psi :D\to \mathbb{R}$, given by

$$\psi \left(x\right)={\int }_0^1\phi (t,x)dt,x\in D,$$

is continuous.

Proof. 

Let $x^0\in D$, and let $\left(x^m\right)\subset D$ be a sequence, such that $x^m\to x^0$ with $m\to \infty$. The set $K=[0,1]\times (\left\{x^0\right\}\cup \{x^m:m\in \mathbb{N}\})$ is compact; so, ${\phi |}_K$ is a uniformly continuous function. Then, for any $\epsilon >0$, there exists $\delta >0$, such that, for any $(t,x),(t,y)\in K$, $\parallel x-y\parallel <\delta$, we get $\left|\phi \right(t,x)-\phi (t,y\left)\right|<\epsilon$. By the fact that $x^m\to x^0$ with $m\to \infty$, there exists $N\in \mathbb{N}$, such that $\parallel x^m-x^0\parallel <\delta$ for all $m>N$. Then, $|\phi (t,x^m)-\phi (t,x^0)|<\epsilon$ for $t\in [0,1]$. So, the sequence ${\left(\phi (t,x^m)\right)}_{m=1}^{\infty }$ is uniformly convergent to $\phi (t,x^0)$ in the interval $[0,1]$.

Finally, we get

$$\psi \left(x^m\right)={\int }_0^1\phi (t,x^m)dt\to {\int }_0^1\phi (t,x^0)dt=\psi \left(x^0\right),\mathrm{with}m\to \infty .$$

 □

Lemma 3.

Let $D\subset {\mathbb{R}}^n$, $G\subset {\mathbb{R}}^{n+1}$ be open sets, such that D is a convex set and $[0,1]\times D\subset G$. Let $\phi :G\to \mathbb{R}$ be a $C^k$-class function, $k⩾1$ ($C^{\infty }$-class function, respectively). Then, the function $\psi :D\to \mathbb{R}$, given by

$$\psi \left(x\right)={\int }_0^1\phi (t,x)dt,x\in D,$$

is a $C^k$-class function ($C^{\infty }$-class function, respectively). In addition,

$${\displaystyle \frac{{\mathsf{\partial }}^{\left|\alpha \right|}\psi }{{\mathsf{\partial }}^{\alpha }x}}\left(x\right)={\int }_0^1{\displaystyle \frac{{\mathsf{\partial }}^{\left|\alpha \right|}\phi }{{\mathsf{\partial }}^{\alpha }x}}(t,x)dt,x=(x_1,\dots ,x_n)\in D$$

(15)

for $\alpha =(i_1,\dots ,i_n)$, $\left|\alpha \right|=i_1+\dots +i_n=1,\dots ,k$($\left|\alpha \right|=1,2,\dots$, respectively).

Proof. 

First, we prove that

$${\displaystyle \frac{\mathsf{\partial }\psi }{\mathsf{\partial }{x}_i}}\left(x\right)={\int }_0^1{\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }{x}_i}}(t,x)dt,x\in D,$$

(16)

for all $i=1,\dots ,n$.

Notice that, it is enough to prove (16) for $i=1$ and $n=1$. Let $x^0\in D$ and $\left(x^m\right)\subset D\setminus \left\{x^0\right\}$ be, such that $x^m\to x^0$ with $m\to \infty$. Then,

$${\displaystyle \frac{\psi \left(x^m\right)-\psi \left(x^0\right)}{x^m-x^0}}={\int }_0^1{\displaystyle \frac{\phi (t,x^m)-\phi (t,x^0)}{x^m-x^0}}dt\mathrm{for}m\in \mathbb{N}.$$

(17)

By the theorem on medium value, we get that, for all $m\in \mathbb{N}$ and $t\in [0,1]$, there exists ${\xi }_{m,t}$ between $x^m$ and $x^0$, such that $\phi (t,x^m)-\phi (t,x^0)=(\mathsf{\partial }\phi /\mathsf{\partial }{x}_1)(t,{\xi }_{m,t})(x^m-x^0)$. So,

$${\displaystyle \frac{\phi (t,x^m)-\phi (t,x^0)}{x^m-x^0}}=(\mathsf{\partial }\phi /\mathsf{\partial }{x}_1)(t,{\xi }_{m,t}).$$

The function $\mathsf{\partial }\phi /\mathsf{\partial }{x}_1$ is continuous, and $|{\xi }_{m,t}-x^0|⩽|x^m-x^0|$. So, similarly to the proof of Lemma 2, we get that the sequence $(\mathsf{\partial }\phi /\mathsf{\partial }{x}_1)(t,{\xi }_{m,t})$ is uniformly convergent to $(\mathsf{\partial }\phi /\mathsf{\partial }{x}_1)(t,x^0)$ in the interval $[0,1]$. Now, taking the limit with $m\to \infty$ in (17), we get (16). Additionally, by Lemma 2, the function $\mathsf{\partial }\psi /\mathsf{\partial }{x}_1$ is continuous.

Finally, using (16), by the induction, we get (15). □

Lemma 4.

Let $D\subset {\mathbb{R}}^n$ be a convex environment of the point $0\in {\mathbb{R}}^n$, $f:D\to \mathbb{R}$ be a $C^k$-class function, $k⩾2$, ($C^{\infty }$-class function, respectively), and $f\left(0\right)=0$, $\mathrm{grad}f\left(0\right)=0$. Let $g_i:D\to \mathbb{R}$, $i=1,\dots ,n$; $h_{i,j}:D\to \mathbb{R}$, $i,j=1,\dots ,n$, be the functions given by

$$\begin{array}{cc}\hfill g_i(x_1,\dots ,x_n)& ={\int }_0^1{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{x}_i}}(t{x}_1,\dots ,t{x}_n)dt,(x_1,\dots ,x_n)\in D,\hfill \\ \hfill h_{i,j}(x_1,\dots ,x_n)& ={\int }_0^1{\displaystyle \frac{\mathsf{\partial }{g}_i}{\mathsf{\partial }{x}_j}}(t{x}_1,\dots ,t{x}_n)dt,(x_1,\dots ,x_n)\in D.\hfill \end{array}$$

Then, $g_i$ are $C^{k-1}$-class functions, $h_{i,j}$ are $C^{k-2}$-class functions ($C^{\infty }$-class functions, respectively),

$$f(x_1,\dots ,x_n)=\sum _{i,j=1}^n{x}_i{x}_j{h}_{i,j}(x_1,\dots ,x_n)\mathit{for}(x_1,\dots ,x_n)\in D,$$

(18)

and

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(0\right)=h_{i,j}\left(0\right).$$

(19)

In particular, the matrix $\left[h_{i,j}\left(0\right)\right]$ is symmetric.

Proof. 

By Lemma 3, the functions $g_i$ are of $C^{k-1}$ class and $h_{i,j}$ are of $C^{k-2}$ class (of $C^{\infty }$ class, respectively), and for any $(x_1,\dots ,x_n)\in D$, we have

$$\begin{array}{cc}\hfill f(x_1,\dots ,x_n)& ={\int }_0^1{\displaystyle \frac{d}{dt}}f(t{x}_1,\dots ,t{x}_n)dt=\sum _{i=1}^n{x}_i{\int }_0^1{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{x}_i}}(t{x}_1,\dots ,t{x}_n)dt\hfill \\ \hfill & =\sum _{i=1}^n{x}_i{g}_i(x_1,\dots ,x_n)=\sum _{i=1}^n{x}_i{\int }_0^1{\displaystyle \frac{d}{dt}}{g}_i(t{x}_1,\dots ,t{x}_n)dt\hfill \\ \hfill & =\sum _{i=1}^n\sum _{j=1}^n{x}_i{x}_j{\int }_0{\displaystyle \frac{\mathsf{\partial }{g}_i}{\mathsf{\partial }{x}_j}}(t{x}_1,\dots ,t{x}_n)dt=\sum _{i,j=1}^n{x}_i{x}_j{h}_{i,j}(x_1,\dots ,x_n),\hfill \end{array}$$

which gives (18). Now, using Lemma 3, we get

$${\displaystyle \frac{\mathsf{\partial }{g}_i}{\mathsf{\partial }{x}_j}}(x_1,\dots ,x_n)={\int }_0^1{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{x}_i}}(t{x}_1,\dots ,t{x}_n)dt={\int }_0^{1}t{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}(t{x}_1,\dots ,t{x}_n)dt.$$

So,

$$h_{i,j}\left(0\right)={\int }_0^1{\displaystyle \frac{\mathsf{\partial }{g}_i}{\mathsf{\partial }{x}_j}}\left(0\right)dt={\displaystyle \frac{\mathsf{\partial }{g}_i}{\mathsf{\partial }{x}_j}}\left(0\right)={\int }_0^{1}t{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(0\right)dt={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(0\right).$$

Thus, $h_{i,j}\left(0\right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(0\right)$, which gives (19). □

Lemma 5.

Let $m⩾n$, and in an environment of the point 0, let

$$f(x_1,\dots ,x_m)=\sum _{i,j=1}^n{x}_i{x}_j{h}_{i,j}(x_1,\dots ,x_m),$$

where $h_{i,j}$ are $C^k$-class functions ($C^{\infty }$-class functions, respectively) in an environment of the point 0, such that

$$h_{i,j}=h_{j,i}fori,j=1,\dots ,n,$$

(20)

$$det{\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,n}\ne 0.$$

(21)

Then, for any $r\in \{1,\dots ,n\}$, there exists a linear exchange of variables $L_r:{\mathbb{R}}^n\to {\mathbb{R}}^n$, such that in an environment of the point 0,

$$f(L_r(x_1,\dots ,x_n),x_{n+1},\dots ,x_m)=\sum _{i,j=1}^n{x}_i{x}_j{\overline{h}}_{i,j}(x_1,\dots ,x_m),$$

(22)

where ${\overline{h}}_{i,j}$ are $C^k$-class functions ($C^{\infty }$-class functions, respectively),

$${\overline{h}}_{i,j}={\overline{h}}_{j,i}fori,j=1,\dots ,n,{\overline{h}}_{r,r}\left(0\right)\ne 0,$$

(23)

and

$$det{\left[{\overline{h}}_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,n}\ne 0.$$

(24)

Proof. 

Fix $r\in \{1,\dots ,n\}$. Assume that there exists $l\in \{1,\dots ,n\}$, such that $h_{l,l}\left(0\right)\ne 0$. Let $L_r=(L_{r,1},\dots ,L_{r,n}):{\mathbb{R}}^n\to {\mathbb{R}}^n$ as follows:

$$L_{r,j}(x_1,\dots ,x_n)=\left\{\begin{array}{cc}{x}_l\hfill & \mathrm{for}j=r,\hfill \\ x_r\hfill & \mathrm{for}j=l,\hfill \\ x_j\hfill & \mathrm{for}j\in \{1,\dots ,n\}\setminus \{r,l\}\hfill \end{array}\right.,$$

and ${\overline{h}}_{i,j}(x_1,\dots ,x_m)=h_{p\left(i\right),p\left(j\right)}(L_r(x_1,\dots ,x_n),x_{n+1},\dots ,x_m)$, where

$$p\left(i\right)=\left\{\begin{array}{cc}l\hfill & \mathrm{for}i=r,\hfill \\ r\hfill & \mathrm{for}i=l,\hfill \\ i\hfill & \mathrm{for}i\in \{1,\dots ,n\}\setminus \{r,l\}.\hfill \end{array}\right.$$

Then, we get the assertion.

Assume now that $h_{l,l}\left(0\right)=0$ for all $l\in \{1,\dots ,n\}$. Then, there exists $i_0\in \{1,\dots ,n\}\setminus \left\{r\right\}$, such that

$$h_{i_0,r}\left(0\right)\ne -h_{r,i_0}\left(0\right).$$

(25)

Otherwise, by (20), we have $h_{l,r}\left(0\right)=0$ for all $l\in \{1,\dots ,n\}$ and $det\left[h_{i,j}\left(0\right)\right]=0$; so, we have a contradiction with (21). Let $i_0\in \{1,\dots ,n\}\setminus \left\{r\right\}$ be such as in (25). Let $L_r=(L_{r,1},\dots ,L_{r,n}):{\mathbb{R}}^n\to {\mathbb{R}}^n$ be as follows:

$$L_{r,j}(x_1,\dots ,x_n)=\left\{\begin{array}{cc}{x}_{i_0}+x_r\hfill & \mathrm{for}j=i_0,\hfill \\ x_j\hfill & \mathrm{for}j\ne i_0.\hfill \end{array}\right.$$

Denote ${\tilde{h}}_{i,j}=h_{i,j}(L_r(x_1,\dots ,x_n),x_{n+1},\dots ,x_m)$ for $i,j=1,\dots ,n$. Then,

$$\begin{array}{cc}\hfill & f(L_r(x_1,\dots ,x_n),x_{n+1},\dots ,x_m)=\sum _{i,j\ne i_0}^n{x}_i{x}_j{\tilde{h}}_{i,j}+{(x_{i_0}+x_r)}^2{\tilde{h}}_{i_0,i_0}\hfill \\ \hfill & +\sum _{i\ne i_0}^n{x}_i(x_{i_0}+x_r){\tilde{h}}_{i,i_0}+\sum _{j\ne i_0}^n(x_{i_0}+x_r)x_j{\tilde{h}}_{i_0,j}\hfill \\ \hfill & =x_r^2[{\tilde{h}}_{r,r}+{\tilde{h}}_{i_0,i_0}+{\tilde{h}}_{r,i_0}+{\tilde{h}}_{i_0,r}]+\sum _{i,j\ne r}^n{x}_i{x}_j{\tilde{h}}_{i,j}\hfill \\ \hfill & +\sum _{j\ne r}^n{x}_r{x}_j[{\tilde{h}}_{r,j}+{\tilde{h}}_{i_0,j}]+\sum _{i\ne r}^n{x}_i{x}_r[{\tilde{h}}_{i,r}+{\tilde{h}}_{i,i_0}].\hfill \end{array}$$

Then, taking

$${\overline{h}}_{i,j}=\left\{\begin{array}{cc}{\tilde{h}}_{r,r}+{\tilde{h}}_{i_0,i_0}+{\tilde{h}}_{r,i_0}+{\tilde{h}}_{i_0,r}\hfill & \mathrm{for}i,j=r\hfill \\ {\tilde{h}}_{i,j}\hfill & \mathrm{for}i,j\ne r\hfill \\ {\tilde{h}}_{r,j}+{\tilde{h}}_{i_0,j}\hfill & \mathrm{for}i=r,j\ne r\hfill \\ {\tilde{h}}_{i,r}+{\tilde{h}}_{i,i_0}\hfill & \mathrm{for}i\ne r,j=r,\hfill \end{array}\right.$$

we get (22) and (23). Condition (24) is a consequence of (21) and the fact that the matrix $\left[{\overline{h}}_{i,j}\right]$ is created of the matrix $\left[{\tilde{h}}_{i,j}\right]$ by adding to r-column of the matrix $\left[{\tilde{h}}_{i,j}\right]$ the $i_0$-column of the matrix $\left[{\tilde{h}}_{i,j}\right]$ and next by adding to r- line of the created matrix the $i_0$-line of this matrix. □

Lemma 6.

Let

$$f(x_1,\dots ,x_n)=\sum _{j=\mathit{r}+1}^n{e}_j{x}_j^2+\sum _{i,j=1}^{\mathit{r}}{x}_i{x}_j{h}_{i,j}(x_1,\dots ,x_n),$$

in an environment of the point $0\in {\mathbb{R}}^n$, $n⩾1$, where $e_j\in \{-1,1\}$, $j=r+1,\dots ,n$, $det{\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}\ne 0$, the matrix ${\left[h_{i,j}\right]}_{i,j=1\dots ,r}$ is symmetric, and $h_{r,r}\left(0\right)\ne 0$ and $h_{i,j}$ are $C^{\infty }$-class functions. Then, there are functions $H_{i,j}(x_1,\dots ,x_n)$, $i,j=1,\dots ,r-1$ of $C^{\infty }$-class in an environment of the point 0, such that

$$f\left(x\right)=\sum _{j=\mathit{r}+1}^n{e}_j{x}_j^2+{\displaystyle \frac{1}{h_{r,r}\left(x\right)}}{\left(\sum _{j=1}^{\mathit{r}}{h}_{j,r}\left(x\right)x_j\right)}^2+\sum _{i,j=1}^{\mathit{r}-1}{H}_{i,j}\left(x\right)x_i{x}_j,$$

(26)

where $x=(x_1,\dots ,x_n)$, $det{\left[H_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r-1}\ne 0$, and $H_{i,j}=H_{j,i}$, $i,j=1,\dots ,r-1$.

Proof. 

Notice that

$$f\left(x\right)=\sum _{j=\mathit{r}+1}^n{e}_j{x}_j^2+\sum _{i,j=1}^{\mathit{r}-1}{h}_{i,j}\left(x\right)x_i{x}_j+R\left(x\right),$$

where

$$\begin{array}{cc}\hfill R\left(x\right)& ={\displaystyle \frac{1}{h_{r,r}\left(x\right)}}\left[2\sum _{j=1}^{\mathit{r}-1}{h}_{j,r}\left(x\right)h_{r,r}\left(x\right)x_j{x}_r+{\left(h_{r,r}\left(x\right)\right)}^2{x}_r^2\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{h_{r,r}\left(x\right)}}\left[{\left(\sum _{j=1}^{\mathit{r}}{h}_{j,r}\left(x\right)x_j\right)}^2-\sum _{i,j=1}^{\mathit{r}-1}{h}_{i,r}\left(x\right)h_{j,r}\left(x\right)x_i{x}_j\right],\hfill \end{array}$$

and $x=(x_1,\dots ,x_n)$. So,

$$\begin{array}{cc}\hfill f\left(x\right)& =\sum _{j=\mathit{r}+1}^n{e}_j{x}_j^2+{\displaystyle \frac{1}{h_{r,r}\left(x\right)}}{\left(\sum _{j=1}^{\mathit{r}}{h}_{j,r}\left(x\right)x_j\right)}^2\hfill \\ \hfill & +\sum _{i,j=1}^{\mathit{r}-1}{x}_i{x}_j\left(h_{i,j}\left(x\right)-{\displaystyle \frac{h_{i,r}\left(x\right)h_{j,r}\left(x\right)}{h_{r,r}\left(x\right)}}\right).\hfill \end{array}$$

Let $H_{i,j}=h_{i,j}-{\displaystyle \frac{h_{i,r}{h}_{j,r}}{h_{r,r}}}$; then, we get (26) and $H_{i,j}=H_{j,i}$ for $i,j=1,\dots ,r-1$. Now, let us complete $r-1$ steps, in succession:

(1) Multiply r-column of the matrix ${\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}$ by ${\displaystyle \frac{h_{1,r}\left(0\right)}{h_{r,r}\left(0\right)}}$ and subtract from the first column to obtain the following matrix:

$$\left(\begin{array}{ccccc}{h}_{1,1}\left(0\right)-{\displaystyle \frac{h_{1,r}\left(0\right)h_{1,r}\left(0\right)}{h_{r,r}\left(0\right)}}& h_{1,2}\left(0\right)& \dots & h_{1,r-1}\left(0\right)& h_{1,r}\left(0\right)\\ ⋮& ⋮& \dots & ⋮& ⋮\\ h_{r-1,1}\left(0\right)-{\displaystyle \frac{h_{r-1,r}\left(0\right)h_{1,r}\left(0\right)}{h_{r,r}\left(0\right)}}& h_{r-1,2}\left(0\right)& \dots & h_{r-1,r-1}\left(0\right)& h_{r-1,r}\left(0\right)\\ h_{r,1}\left(0\right)-{\displaystyle \frac{h_{r,r}\left(0\right)h_{1,r}\left(0\right)}{h_{r,r}\left(0\right)}}& h_{r,2}\left(0\right)& \dots & h_{r,r-1}\left(0\right)& h_{r,r}\left(0\right)\end{array}\right);$$

(2) Multiply r-column of the matrix ${\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}$ by ${\displaystyle \frac{h_{2,r}\left(0\right)}{h_{r,r}\left(0\right)}}$, subtract from the second column, and use the symmetry of the matrix ${\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}$, to obtain the following matrix:

$$\left(\begin{array}{ccccc}{H}_{1,1}\left(0\right)& h_{1,2}\left(0\right)-{\displaystyle \frac{h_{1,r}\left(0\right)h_{2,r}\left(0\right)}{h_{r,r}\left(0\right)}}& \dots & h_{1,r-1}\left(0\right)& h_{1,r}\left(0\right)\\ ⋮& ⋮& \dots & ⋮& ⋮\\ H_{r-1,1}\left(0\right)& h_{r-1,2}\left(0\right)-{\displaystyle \frac{h_{r-1,r}\left(0\right)h_{2,r}\left(0\right)}{h_{r,r}\left(0\right)}}& \dots & h_{r-1,r-1}\left(0\right)& h_{r-1,r}\left(0\right)\\ 0& h_{r,2}\left(0\right)-{\displaystyle \frac{h_{r,r}\left(0\right)h_{2,r}\left(0\right)}{h_{r,r}\left(0\right)}}& \dots & h_{r,r-1}\left(0\right)& h_{r,r}\left(0\right)\end{array}\right);$$

…,

($r-1$) multiply r- column of the matrix ${\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}$ by ${\displaystyle \frac{h_{r-1,r}\left(0\right)}{h_{r,r}\left(0\right)}}$, subtract from $(r-1)$- column and use the symmetry of the matrix ${\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}$, to obtain the following matrix:

$$\left(\begin{array}{ccccc}{H}_{1,1}\left(0\right)& H_{1,2}\left(0\right)& \dots & h_{1,r-1}\left(0\right)-{\displaystyle \frac{h_{1,r}\left(0\right)h_{r-1,r}\left(0\right)}{h_{r,r}\left(0\right)}}& h_{1,r}\left(0\right)\\ ⋮& ⋮& \dots & ⋮& ⋮\\ H_{r-1,1}\left(0\right)& H_{r-1,2}\left(0\right)& \dots & h_{r-1,r-1}\left(0\right)-{\displaystyle \frac{h_{r-1,r}\left(0\right)h_{r-1,r}\left(0\right)}{h_{r,r}\left(0\right)}}& h_{r-1,r}\left(0\right)\\ 0& 0& \dots & h_{r,r-1}\left(0\right)-{\displaystyle \frac{h_{r,r}\left(0\right)h_{r-1,r}\left(0\right)}{h_{r,r}\left(0\right)}}& h_{r,r}\left(0\right)\end{array}\right).$$

Next, again by the symmetry of the matrix ${\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}$, we finally obtain the following matrix:

$$\left(\begin{array}{ccccc}{H}_{1,1}\left(0\right)& H_{1,2}\left(0\right)& \dots & H_{1,r-1}\left(0\right)& h_{1,r}\left(0\right)\\ ⋮& ⋮& \dots & ⋮& ⋮\\ H_{r-1,1}\left(0\right)& H_{r-1,2}\left(0\right)& \dots & H_{r-1,r-1}\left(0\right)& h_{r-1,r}\left(0\right)\\ 0& 0& \dots & 0& h_{r,r}\left(0\right)\end{array}\right).$$

So, we get

$$\begin{array}{cc}\hfill det{\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}& =h_{r,r}\left(0\right){(-1)}^{2r}det{\left[H_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r-1}\hfill \\ \hfill & =h_{r,r}\left(0\right)det{\left[H_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r-1}.\hfill \end{array}$$

By the assumption, we have $h_{r,r}\left(0\right)\ne 0$ and $det{\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}\ne 0$. Therefore, $det{\left[H_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r-1}\ne 0$. □

Now, we are going to prove Theorem 10.

Proof. Morse Lemma.

Assume that $a=0\in {\mathbb{R}}^n$. By Lemma 4, we have

$$f(x_1,\dots ,x_n)=\sum _{i,j=1}^n{x}_i{x}_j{h}_{i,j}(x_1,\dots ,x_n),$$

in a certain environment of $0\in {\mathbb{R}}^n$, where $h_{i,j}$ are $C^{\infty }$-class functions,

$$h_{i,j}=h_{j,i},{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(0\right)=h_{i,j}\left(0\right)fori,j=1,\dots ,n.$$

So,

$$det{\left[h_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,n}={\displaystyle \frac{1}{2^n}}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(0\right)\right]\ne 0.$$

Let us now prove that, for any $r=0,\dots ,n$, there exists a diffeomorphism ${\phi }_r:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$ of $C^{\infty }$-class, such that, in a certain environment of $0\in {\mathbb{R}}^n$, we have

$$f\circ {\phi }_r\left(x\right)=\sum _{j=\mathit{r}+1}^n{e}_j{x}_j^2+\sum _{i,j=1}^{\mathit{r}}{\overline{H}}_{i,j}\left(x\right)x_i{x}_j,$$

(27)

where $e_{r+1},\dots ,e_n\in \{-1,1\}$, ${\overline{H}}_{i,j}$, $i,j=1,\dots ,r$ are $C^{\infty }$-class functions, ${\overline{H}}_{i,j}={\overline{H}}_{j,i}$, and $det{\left[{\overline{H}}_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}\ne 0$.

Let use the induction by $n-r$. For $n-r=0$, we have $r=n$; so, by Lemma 4, we get (27). Assume now that (27) holds for $n-r-1⩾0$. We are going to show that (27) holds for $n-r$. By the assumption of induction, there exists a diffeomorphism ${\phi }_{r+1}:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$, such that, in a certain environment of $0\in {\mathbb{R}}^n$, we have

$$f\circ {\phi }_{r+1}\left(x\right)=\sum _{j=\mathit{r}+2}^n{e}_j{x}_j^2+\sum _{i,j=1}^{\mathit{r}+1}{x}_i{x}_j{H}_{i,j}\left(x\right),$$

where $e_{r+2},\dots ,e_n\in \{-1,1\}$, $H_{i,j}$ are $C^{\infty }$-class functions, $H_{i,j}=H_{j,i}$, $i,j=1,\dots ,r+1$, and $det{\left[H_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r+1}\ne 0$. Using Lemma 5, we can assume that $H_{r+1,r+1}\left(0\right)\ne 0$. Then, by Lemma 6, there exist some functions ${\overline{H}}_{i,j}$ of $C^{\infty }$-class, $i,j=1,\dots ,r$, such that

$$f\circ {\phi }_{r+1}\left(x\right)=\sum _{j=\mathit{r}+2}^n{e}_j{x}_j^2+{\displaystyle \frac{1}{H_{r+1,r+1}\left(x\right)}}{\left(\sum _{j=1}^{\mathit{r}+1}{H}_{j,r+1}\left(x\right)x_j\right)}^2+\sum _{i,j=1}^{\mathit{r}}{\overline{H}}_{i,j}\left(x\right)x_i{x}_j,$$

where $x=(x_1,\dots ,x_n)$, ${\overline{H}}_{i,j}={\overline{H}}_{j,i}$ for $i,j=1,\dots ,r$, and $det{\left[{\overline{H}}_{i,j}\left(0\right)\right]}_{i,j=1,\dots ,r}\ne 0$. Let $g:({\mathbb{R}}^n,0)\to \mathbb{R}$, $g\left(y\right)=\sqrt{\mid H_{r+1,r+1}\left(y\right)\mid }$, be the function defined in a certain environment of $0\in {\mathbb{R}}^n$. Taking

$$\begin{array}{c}\hfill \psi \left(y\right)=({\psi }_1\left(y\right),\dots ,{\psi }_n\left(y\right))=(y_1,\dots ,y_r,{\displaystyle \frac{1}{g\left(y\right)}}\sum _{j=1}^{\mathit{r}+1}{H}_{j,r+1}\left(y\right)y_j,y_{r+2},\dots ,y_n),\end{array}$$

we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\partial }{\psi }_{r+1}}{\mathsf{\partial }{y}_i}}\left(0\right)& ={\displaystyle \frac{\mathsf{\partial }\left(\frac{1}{g\left(y\right)}{\sum }_{j=1}^{\mathit{r}+1}{H}_{j,r+1}\left(y\right)y_j\right)}{\mathsf{\partial }{y}_i}}\left(0\right)={\displaystyle \frac{1}{g\left(0\right)}}{\displaystyle \frac{\mathsf{\partial }\left({\sum }_{j=1}^{\mathit{r}+1}{H}_{j,r+1}\left(y\right)y_j\right)}{\mathsf{\partial }{y}_i}}\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{g\left(0\right)}}{H}_{i,r+1}\left(0\right)\hfill & \mathrm{for}i\in \{1,\dots ,r+1\},\hfill \\ 0\hfill & \mathrm{for}i\in \{r+2,\dots ,n\}.\hfill \end{array}\right.\hfill \end{array}$$

So,

$$det{\left[{\displaystyle \frac{\mathsf{\partial }{\psi }_i}{\mathsf{\partial }{y}_j}}\left(0\right)\right]}_{i,j=1,\dots ,n}={\displaystyle \frac{H_{r+1,r+1}\left(0\right)}{g\left(0\right)}}=eg\left(0\right),$$

where $e\in \{-1,1\}$. Therefore, $det{\left[{\displaystyle \frac{\mathsf{\partial }{\psi }_i}{\mathsf{\partial }{y}_j}}\left(0\right)\right]}_{i,j=1,\dots ,n}\ne 0$, and then, $\psi :({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$ is a diffeomorphism. Let ${\overline{\phi }}_r={\psi }^{-1}$. Then, ${\overline{\phi }}_r:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$ is a diffeomorphism, and for a certain function ${\xi }_{r+1}:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$ of $C^{\infty }$-class, the following holds:

$${\overline{\phi }}_r\left(x\right)=(x_1,\dots ,x_r,{\xi }_{r+1}\left(x\right),x_{r+2},\dots ,x_n).$$

We denote $e_{r+1}=\mathrm{sgn}\left(H_{r+1,r+1}\left(0\right)\right)$. Then,

$$\begin{array}{cc}\hfill & f\circ {\phi }_{r+1}\circ {\overline{\phi }}_r\left(x\right)=f\circ {\phi }_{r+1}(x_1,\dots ,x_r,{\xi }_{r+1}\left(x\right),x_{r+2},\dots ,x_n)=\sum _{j=\mathit{r}+2}^n{e}_j{x}_j^2\hfill \\ \hfill & +e_{r+1}{\left[{\displaystyle \frac{1}{g\circ {\overline{\phi }}_r\left(x\right)}}\left(\sum _{j=1}^{\mathit{r}}{H}_{j,r+1}\circ {\overline{\phi }}_r\left(x\right)x_j+H_{r+1,r+1}\circ {\overline{\phi }}_r\left(x\right){\xi }_{r+1}\left(x\right)\right)\right]}^2\hfill \\ \hfill & +\sum _{i,j=1}^{\mathit{r}}{\overline{H}}_{i,j}\circ {\overline{\phi }}_r\left(x\right)x_i{x}_j=\sum _{j=\mathit{r}+2}^n{e}_j{x}_j^2+e_{r+1}{\left({\psi }_{r+1}\circ {\psi }^{-1}\left(x\right)\right)}^2\hfill \\ \hfill & +\sum _{i,j=1}^{\mathit{r}}{\overline{H}}_{i,j}\circ {\overline{\phi }}_r\left(x\right)x_i{x}_j=\sum _{j=\mathit{r}+2}^n{e}_j{x}_j^2+e_{r+1}{x}_{r+1}^2+\sum _{i,j=1}^{\mathit{r}}{\overline{H}}_{i,j}\circ {\overline{\phi }}_r\left(x\right)x_i{x}_j\hfill \\ \hfill & =\sum _{j=\mathit{r}+1}^n{e}_j{x}_j^2+\sum _{i,j=1}^{\mathit{r}}{\overline{H}}_{i,j}\circ {\overline{\phi }}_r\left(x\right)x_i{x}_j.\hfill \end{array}$$

Putting ${\phi }_r={\phi }_{r+1}\circ {\overline{\phi }}_r$, we get (27), for $n-r$. The induction finishes the proof of (27).

Let $\overline{\phi }={\phi }_0$. Then, $\overline{\phi }:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$ is a diffeomorphism, and

$$f\circ \overline{\phi }\left(x\right)=\sum _{j=1}^n{e}_j{x}_j^2,$$

in a certain environment of $0\in {\mathbb{R}}^n$, where $e_1,\dots ,e_n\in \{-1,1\}$.

Let $l=\#\{i\in \{1,\dots ,n\}:e_i=-1\}$. Then, $\#\{i\in \{1,\dots ,n\}:e_i=1\}=n-l$. Let $\{i\in \{1,\dots ,n\}:e_i=-1\}=\{i_1,\dots ,i_l\}$, $\{i\in \{1,\dots ,n\}:e_i=1\}=\{i_{l+1},\dots ,i_n\}$. We denote by $\sigma$ the permutation of the set $\{1,\dots ,n\}$, such that

$$\sigma \left(i_j\right)=j$$

for $j\in \{1,\dots ,n\}$.

Define the linear mapping $L:{\mathbb{R}}^n\to {\mathbb{R}}^n$ by

$$L(x_1,\dots ,x_n)=(x_{\sigma \left(1\right)},\dots ,x_{\sigma \left(n\right)})$$

for $(x_1,\dots ,x_n)\in {\mathbb{R}}^n$.

Taking $\phi =\overline{\phi }\circ L$, we get the thesis of the Morse Lemma. □

Notice that the Morse Lemma is also valid for $n=1$:

Proposition 8.

Let f be a $C^{\infty }$-class function in an environment of the point $a\in \mathbb{R}$, such that

$$f\left(a\right)=0,f^{\prime }\left(a\right)=0,f^{″}\left(a\right)\ne 0.$$

Then, there is a diffeomorphism $\phi :(\mathbb{R},a)\to (\mathbb{R},a)$ of $C^{\infty }$-class, such that

$$f\circ \phi \left(x\right)=e{x}^2\mathit{in}\mathit{a}\mathit{certain}\mathit{environment}\mathit{of}a,$$

where $e\in \{-1,1\}$.

Proof. 

We can assume that $a=0\in \mathbb{R}$. Using Lemma 4, we get that, in a certain environment of $a=0\in \mathbb{R}$, the following holds:

$$f\left(x\right)=x^{2}h\left(x\right),$$

where h is a $C^{\infty }$-class function, and

$$\begin{array}{cc}\hfill 0\ne f^{″}\left(0\right)& ={\left[x^{2}h\left(x\right)\right]}^{″}\left(0\right)={[2xh\left(x\right)+x^2{h}^{\prime }\left(x\right)]}^{\prime }\left(0\right)\hfill \\ \hfill & =[2h\left(x\right)+2x{h}^{\prime }\left(x\right)+2x{h}^{\prime }\left(x\right)+x^2{h}^{″}\left(x\right)]\left(0\right)=2h\left(0\right).\hfill \end{array}$$

So, $h\left(0\right)\ne 0$.

Let $e=\mathrm{sgn}\left(h\right(0\left)\right)$. Then, $e\in \{-1,1\}$, and in a certain environment of $a=0\in \mathbb{R}$, we have

$$h\left(x\right)=e{\left(\sqrt{\mid h\left(x\right)\mid }\right)}^2.$$

So,

$$f\left(x\right)=e{\left(x\sqrt{\left|h\right(x\left)\right|}\right)}^2$$

in a certain environment of $a=0\in \mathbb{R}$. Let

$$\psi \left(x\right)=x\sqrt{\left|h\right(x\left)\right|}.$$

Then, $\psi$ is of $C^{\infty }$-class, $\psi \left(0\right)=0$, and

$${\psi }^{\prime }\left(0\right)={\left(x\sqrt{\left|h\right(x\left)\right|}\right)}^{\prime }\left(0\right)=\left[\sqrt{\left|h\right(x\left)\right|}+x{\displaystyle \frac{h^{\prime }\left(x\right)}{2\sqrt{\left|h\right(x\left)\right|}}}\right]\left(0\right)=\sqrt{\left|h\right(0\left)\right|}>0.$$

So, $\psi :(\mathbb{R},0)\to (\mathbb{R},0)$ is a diffeomorphism of $C^{\infty }$-class. Let $\phi ={\psi }^{-1}$. Then, $\phi :(\mathbb{R},0)\to (\mathbb{R},0)$ is a diffeomorphism of $C^{\infty }$-class, and in a certain environment of $a=0\in \mathbb{R}$, the following holds:

$$f\circ \phi \left(x\right)=\left(e{\psi }^2\right)\circ \phi \left(x\right)=e{[\psi \circ \phi \left(x\right)]}^2=e{[\psi \circ {\psi }^{-1}\left(x\right)]}^2=e{x}^2.$$

 □

Let f be a $C^{\infty }$-class function in an environment of the point $a\in {\mathbb{R}}^n$. The point a is called the non-degenerate critical point of f, when

$$\mathrm{grad}f\left(a\right)=0,\mathrm{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

By Theorem 10 and Proposition 8, we get immediately (see also [2]):

Corollary 3.

The set of the non-degenerate critical points of a $C^{\infty }$-class function specified in an open set in ${\mathbb{R}}^n$ is isolated.

4.2. Morse Lemma—The Complex Case

Let us now introduce the Morse Lemma for a holomorphic function $f:G\to \mathbb{C}$, where $G\subset {\mathbb{C}}^n$ is an open set. The gradient of f, as usual, is defined by

$$\mathrm{grad}f=\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{z}_1}},\dots ,{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{z}_n}}\right):G\to {\mathbb{C}}^n.$$

Theorem 11. Morse Lemma for a holomorphic function.

Let f be a holomorphic function in a certain environment of the point $\stackrel{\mathrm{o}}{z}=({\stackrel{\mathrm{o}}{z}}_1,\dots ,{\stackrel{\mathrm{o}}{z}}_n)\in {\mathbb{C}}^n$, $n⩾1$, such that

$$f\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathrm{grad}f\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{z}_i\mathsf{\partial }{z}_j}}\left(\stackrel{\mathrm{o}}{z}\right)\right]\ne 0.$$

(28)

Then, there exists a biholomorphism φ in a certain environment of $\stackrel{\mathrm{o}}{z}$, such that

$$f\circ \phi \left(z\right)=\sum _{j=1}^n{(z_j-{\stackrel{\mathrm{o}}{z}}_j)}^2.$$

First, we give two lemmas necessary for the proof of Theorem 11.

By analogy, as in Lemma 5, we prove the following:

Lemma 7.

Let $m⩾n$,

$$f(z_1,\dots ,z_m)=\sum _{l,j=1}^n{z}_l{z}_j{h}_{l,j}(z_1,\dots ,z_m),$$

in an environment of the point $0\in {\mathbb{C}}^m$, where $h_{l,j}$ are holomorphic functions, $h_{l,j}=h_{j,l}$ for $l,j=1,\dots ,n$, and $det{\left[h_{l,j}\left(0\right)\right]}_{l,j=1,\dots ,n}\ne 0$. Then, for any $r\in \{1,\dots ,n\}$, there exists a linear exchange of variables $L_r:{\mathbb{C}}^n\to {\mathbb{C}}^n$, such that, in an environment of 0,

$$f(L_r(z_1,\dots ,z_n),z_{n+1},\dots ,z_m)=\sum _{l,j=1}^n{z}_l{z}_j{\overline{h}}_{l,j}(z_1,\dots ,z_m),$$

where ${\overline{h}}_{l,j}$ are holomorphic functions, ${\overline{h}}_{l,j}={\overline{h}}_{j,l}$ for $l,j=1,\dots ,n$, ${\overline{h}}_{r,r}\left(0\right)\ne 0$ and $det{\left[{\overline{h}}_{l,j}\left(0\right)\right]}_{l,j=1,\dots ,n}\ne 0$.

By analogy, as in Lemma 6, we prove the following:

Lemma 8.

Let, in an environment of the point $0\in {\mathbb{C}}^n$, $n⩾1$,

$$f(z_1,\dots ,z_n)=\sum _{j=\mathit{r}+1}^n{z}_j^2+\sum _{l,j=1}^{\mathit{r}}{z}_l{z}_j{h}_{l,j}(z_1,\dots ,z_n),$$

where $h_{l,j}$ are holomorphic functions, $det{\left[h_{l,j}\left(0\right)\right]}_{l,j=1,\dots ,r}\ne 0$, ${\left[h_{l,j}\right]}_{l,j=1\dots ,r}$ is a symmetric matrix, and $h_{r,r}\left(0\right)\ne 0$. Then, there are holomorphic (in an environment of 0) functions $H_{l,j}(z_1,\dots ,z_n)$, $l,j=1,\dots ,r-1$, such that

$$f\left(z\right)=\sum _{j=\mathit{r}+1}^n{z}_j^2+{\displaystyle \frac{1}{h_{r,r}\left(z\right)}}{\left(\sum _{j=1}^{\mathit{r}}{h}_{j,r}\left(z\right)z_j\right)}^2+\sum _{l,j=1}^{\mathit{r}-1}{H}_{l,j}\left(z\right)z_l{z}_j,$$

where $z=(z_1,\dots ,z_n)$, $det{\left[H_{l,j}\left(0\right)\right]}_{l,j=1,\dots ,r-1}\ne 0$, and $H_{l,j}=H_{j,l}$, $l,j=1,\dots ,$$r-1$.

Now, we can prove the main theorem.

Proof. Morse Lemma for a holomorphic function.

Assume that $\stackrel{\mathrm{o}}{z}=0\in {\mathbb{C}}^n$. Let P be a policylinder with center at $0\in {\mathbb{C}}^n$, such that (28) holds in P. Then, by Lemma 1, there exist holomorphic functions $f_j:P\to \mathbb{C}$, $j=1,\dots ,n$, such that, for $z\in P$, we have

$$f\left(z\right)=\sum _{j=1}^n{z}_j{f}_j\left(z\right).$$

Additionally, by Theorem 2$\left(a\right)\Rightarrow \left(b\right)$ and by Proposition 2, we get

$$f_j\left(0\right)={\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{z}_j}}\left(0\right)=0,j=1,\dots ,n.$$

Using again Lemma 1 to functions $f_j$, $j=1,\dots ,n$, we get that there exist holomorphic functions ${\overline{h}}_{j,l}:P\to \mathbb{C}$, $j,l=1,\dots ,n$, such that

$$f_j\left(z\right)=\sum _{l=1}^n{z}_l{\overline{h}}_{j,l}\left(z\right),j=1,\dots ,n$$

for $z\in P$. Additionally, by Theorem 2$\left(a\right)\Rightarrow \left(b\right)$ and by Proposition 2, the following holds:

$${\displaystyle \frac{\mathsf{\partial }{f}_j}{\mathsf{\partial }{z}_l}}\left(0\right)={\overline{h}}_{j,l}\left(0\right),j,l=1,\dots ,n.$$

We have

$$f\left(z\right)=\sum _{j=1}^n{z}_j{f}_j\left(z\right)=\sum _{j\ne l}^n{z}_j{f}_j\left(z\right)+z_l{f}_l\left(z\right).$$

So,

$${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{z}_l}}\left(z\right)=\sum _{j\ne l}^n{z}_j{\displaystyle \frac{\mathsf{\partial }{f}_j}{\mathsf{\partial }{z}_l}}\left(z\right)+f_l\left(z\right)+z_l{\displaystyle \frac{\mathsf{\partial }{f}_l}{\mathsf{\partial }{z}_l}}\left(z\right).$$

Therefore,

$${\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{z}_l\mathsf{\partial }{z}_k}}\left(0\right)={\displaystyle \frac{\mathsf{\partial }{f}_k}{\mathsf{\partial }{z}_l}}\left(0\right)+{\displaystyle \frac{\mathsf{\partial }{f}_l}{\mathsf{\partial }{z}_k}}\left(0\right)={\overline{h}}_{k,l}\left(0\right)+{\overline{h}}_{l,k}\left(0\right)\mathrm{for}k,l=1,\dots ,n.$$

Let $h_{j,l}={\displaystyle \frac{1}{2}}({\overline{h}}_{j,l}+{\overline{h}}_{l,j})$, $j,l=1,\dots ,n$. Then,

$$f\left(z\right)=\sum _{j,l=1}^n{z}_j{z}_l{h}_{j,l}\left(z\right)\mathrm{for}z\in P,$$

(29)

where $h_{j,l}$ are holomorphic functions, $h_{j,l}=h_{l,j}$ and ${\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{z}_l\mathsf{\partial }{z}_j}}\left(0\right)=h_{j,l}\left(0\right)$ for $j,l=1,\dots ,n$.

Now, we prove that, for any $k\in \{0,1,\dots ,n\}$, there exists a biholomorphism ${\phi }_k:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$, such that

$$f\circ {\phi }_k\left(z\right)=\sum _{j=\mathit{k}+1}^n{z}_j^2+\sum _{j,l=1}^{\mathit{k}}{\overline{H}}_{j,l}\left(z\right)z_j{z}_l,$$

(30)

in a certain environment of $0\in {\mathbb{C}}^n$, where ${\overline{H}}_{j,l}$ are holomorphic functions, ${\overline{H}}_{j,l}={\overline{H}}_{l,j}$, $j,l=1,\dots ,k$, and $det{\left[{\overline{H}}_{j,l}\left(0\right)\right]}_{j,l=1,\dots ,k}\ne 0$.

Let use the induction by $n-k$. For $n-k=0$, we have $k=n$. So, by (29), we get (30). Assume now that (30) holds for $n-k-1⩾0$. We are going to show that (30) holds for $n-k$. By the assumption of induction, there exists a biholomorphism ${\phi }_{k+1}:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$, such that, in a certain environment of $0\in {\mathbb{C}}^n$, we have

$$f\circ {\phi }_{k+1}\left(z\right)=\sum _{j=\mathit{k}+2}^n{z}_j^2+\sum _{j,l=1}^{\mathit{k}+1}{H}_{j,l}\left(z\right)z_j{z}_l,$$

(31)

where $H_{j,l}$ are holomorphic functions, $H_{j,l}=H_{l,j}$ for $j,l=1,\dots ,k+1$, and $det{\left[H_{j,l}\left(0\right)\right]}_{j,l=1,\dots ,k+1}$$\ne 0$. By Lemma 7, we can assume that $H_{k+1,k+1}\left(0\right)\ne 0$. Let $\Omega$ be a policylinder with center at $0\in {\mathbb{C}}^n$, such that (31) holds in $\Omega$. Then, by Theorem 4, there exists a holomorphic function $L:\Omega \to \mathbb{C}$, such that

$$exp\circ L=H_{k+1,k+1}.$$

So, L is a branch of logarithm of $H_{k+1,k+1}$. Let

$$P=exp\left[{\displaystyle \frac{1}{2}}L\left(z\right)\right].$$

Then, P is a holomorphic function in $\Omega$, such that

$$P^2\left(z\right)=H_{k+1,k+1}\left(z\right),P\left(z\right)\ne 0,and$$

$${\displaystyle \frac{1}{H_{k+1,k+1}\left(z\right)}}={\left({\displaystyle \frac{1}{P\left(z\right)}}\right)}^{2}forz\in \Omega .$$

Let $\psi =({\psi }_1,\dots ,{\psi }_n)$ be a mapping given by

$$\psi \left(w\right)=(w_1,\dots ,w_k,{\displaystyle \frac{1}{P\left(w\right)}}\left(\sum _{j=1}^{\mathit{k}+1}{H}_{j,k+1}\left(w\right)w_j\right),w_{k+2},\dots ,w_n).$$

Therefore, $\psi$ is a holomorphic mapping, and by analogy, as in Theorem 10 we can show that $det\left[{\displaystyle \frac{\mathsf{\partial }{\psi }_l}{\mathsf{\partial }{w}_j}}\left(0\right)\right]\ne 0$. So, by Theorem 3, $\psi :({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ is a biholomorphism. Let ${\overline{\phi }}_k={\psi }^{-1}$. Then, ${\overline{\phi }}_k$ is a biholomorphic mapping, and for a certain holomorphic function ${\xi }_{k+1}$, the following holds:

$${\overline{\phi }}_k\left(z\right)=(z_1,\dots ,z_k,{\xi }_{k+1}\left(z\right),z_{k+2},\dots ,z_n),$$

in an environment of zero. Now, using Lemma 8, we get that there exist holomorphic functions ${\overline{H}}_{l,j}$, $j,l=1,\dots ,k$, such that, in a certain environment of $0\in {\mathbb{C}}^n$, the following holds:

$$f\circ {\phi }_{k+1}\left(z\right)=\sum _{j=\mathit{k}+2}^n{z}_j^2+{\displaystyle \frac{1}{H_{k+1,k+1}\left(z\right)}}{\left(\sum _{j=1}^{\mathit{k}+1}{H}_{j,k+1}\left(z\right)z_j\right)}^2+\sum _{j,l=1}^{\mathit{k}}{\overline{H}}_{j,l}\left(z\right)z_j{z}_l,$$

where ${\overline{H}}_{j,l}={\overline{H}}_{l.j}$ for $j,l=1,\dots ,k$, and $det{\left[{\overline{H}}_{j,l}\left(0\right)\right]}_{j,l=1,\dots ,k}\ne 0$. So, in a certain environment of zero, we get

$$\begin{array}{cc}\hfill & f\circ {\phi }_{k+1}\circ {\overline{\phi }}_k\left(z\right)=f\circ {\phi }_{k+1}(z_1,\dots ,z_k,{\xi }_{k+1}\left(z\right),z_{k+2},\dots ,z_n)\hfill \\ \hfill & =\sum _{j=\mathit{k}+2}^n{z}_j^2+{\displaystyle \frac{1}{{\left(P\left({\overline{\phi }}_k\left(z\right)\right)\right)}^2}}{\left(\sum _{j=1}^{\mathit{k}}{H}_{j,k+1}\left({\overline{\phi }}_k\left(z\right)\right)z_j+H_{k+1,k+1}\left({\overline{\phi }}_k\left(z\right)\right){\xi }_{k+1}\left(z\right)\right)}^2\hfill \\ \hfill & +\sum _{j,l=1}^k{\overline{H}}_{j,l}\left({\overline{\phi }}_k\left(z\right)\right)z_j{z}_l=\sum _{j=\mathit{k}+2}^n{z}_j^2+{\left({\psi }_{k+1}\left({\overline{\phi }}_k\left(z\right)\right)\right)}^2+\sum _{j,l=1}^{\mathit{k}}{\overline{H}}_{j,l}\left({\overline{\phi }}_k\left(z\right)\right)z_j{z}_l\hfill \\ \hfill & =\sum _{j=\mathit{k}+2}^n{z}_j^2+{\left({\psi }_{k+1}\left({\psi }^{-1}\left(z\right)\right)\right)}^2+\sum _{j,l=1}^{\mathit{k}}{\overline{H}}_{j,l}\left({\overline{\phi }}_k\left(z\right)\right)z_j{z}_l=\sum _{j=\mathit{k}+2}^n{z}_j^2+z_{k+1}^2\hfill \\ \hfill & +\sum _{j,l=1}^{\mathit{k}}{\overline{H}}_{j,l}\left({\overline{\phi }}_k\left(z\right)\right)z_j{z}_l=\sum _{j=\mathit{k}+1}^n{z}_j^2+\sum _{j,l=1}^{\mathit{k}}{\overline{H}}_{j,l}\left({\overline{\phi }}_k\left(z\right)\right)z_j{z}_l.\hfill \end{array}$$

Taking ${\phi }_k={\phi }_{k+1}\circ {\overline{\phi }}_k$ we get (30) for $n-k$. The induction finishes the proof of (30).

Finally, let $\phi ={\phi }_0$. Then, $\phi :({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ is a biholomorphism, and

$$f\circ \phi \left(z\right)=\sum _{j=1}^n{z}_j^2,$$

in an environment of $0\in {\mathbb{C}}^n$. □

Corollary 4.

Let f be a holomorphic function in a certain environment of the point $\stackrel{\mathrm{o}}{z}\in \mathbb{C}$, such that

$$f\left(\stackrel{\mathrm{o}}{z}\right)=0,f^{\prime }\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathit{and}{f}^{″}\left(\stackrel{\mathrm{o}}{z}\right)\ne 0.$$

Then, there exists a biholomorphism φ in a certain environment of the point $\stackrel{\mathrm{o}}{z}$, such that

1. 

The Laurent series of $f\circ \phi$ is of the form ${\sum }_{n=-\infty }^{+\infty }{a}_n{(z-\stackrel{\mathrm{o}}{z})}^n={(z-\stackrel{\mathrm{o}}{z})}^2$ in a certain environment of $\stackrel{\mathrm{o}}{z}$;

2. 

$\stackrel{\mathrm{o}}{z}$ is an apparent critical point of $f\circ \phi$;

3. 

$f\circ \phi$ is meromorphic, and ${\mathrm{ord}}_{\stackrel{\mathrm{o}}{z}}(f\circ \phi )=2$;

4. 

$\stackrel{\mathrm{o}}{z}$ is a $k=2$ root of $f\circ \phi$;

5. 

$1/(f\circ \phi )$ is meromorphic and $\stackrel{\mathrm{o}}{z}$ is a $k=2$ pole of $1/(f\circ \phi )$.

Proof. 

This is a direct consequence of Theorem 11, Corollary 1, and Corollary 2. □

Corollary 5.

Let $f,g$ be holomorphic functions in certain environments of the point $\stackrel{\mathrm{o}}{z}\in {\mathbb{C}}^n$, $n⩾1,$ such that

$$f\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathrm{grad}f\left(\stackrel{\mathrm{o}}{z}\right)=0,det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{z}_l\mathsf{\partial }{z}_k}}\left(\stackrel{\mathrm{o}}{z}\right)\right]\ne 0,$$

and

$$g\left(\stackrel{\mathrm{o}}{z}\right)=0,\mathrm{grad}g\left(\stackrel{\mathrm{o}}{z}\right)=0,det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}g}{\mathsf{\partial }{z}_l\mathsf{\partial }{z}_k}}\left(\stackrel{\mathrm{o}}{z}\right)\right]\ne 0.$$

Then, there exists an environment U of the point $\stackrel{\mathrm{o}}{z}$ and a biholomorphism $\phi :({\mathbb{C}}^n,\stackrel{\mathrm{o}}{z})\to ({\mathbb{C}}^n,\stackrel{\mathrm{o}}{z})$, such that

$$f\left(z\right)=(g\circ \phi )\left(z\right)z\in U.$$

Proof. 

Using Theorem 11, we get that there exist biholomorphisms ${\phi }_1,{\phi }_2$ in certain environments of the point $\stackrel{\mathrm{o}}{z}$, such that

$$f\circ {\phi }_1\left(z\right)=\sum _{k=1}^n{(z_k-{\stackrel{\mathrm{o}}{z}}_k)}^2\mathrm{in}\mathrm{a}\mathrm{certain}\mathrm{environment}\mathrm{of}\stackrel{\mathrm{o}}{z},$$

and

$$g\circ {\phi }_2\left(z\right)=\sum _{k=1}^n{(z_k-{\stackrel{\mathrm{o}}{z}}_k)}^2\mathrm{in}\mathrm{a}\mathrm{certain}\mathrm{environment}\mathrm{of}\stackrel{\mathrm{o}}{z}.$$

Taking $\phi ={\phi }_2\circ {\phi }_1^{-1}$ and as U the intersection of the ${\phi }_2$ and ${\phi }_1^{-1}$ domains, we attain the assertion. □

5. The Index of a Function in a Non-Degenerate Critical Point

Let $F:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ be a bilinear symmetric form, where

$$\Gamma =\{V\subset {\mathbb{R}}^n:\mathrm{V}\mathrm{is}\mathrm{a}\mathrm{linear}\mathrm{subspace},F(v,v)<0\mathrm{for}v\in V\setminus \left\{0\right\}\}.$$

The index of a form F is the integer number $l\in \{0,1,\dots ,n\}$, such that

$$l=\text{max}\{\dim V:V\in \Gamma \}.$$

Let f be a $C^{\infty }$-class function in an environment of a point $a\in {\mathbb{R}}^n$, such that $\mathrm{grad}f\left(a\right)=0$. Let $A={\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]}_{i,j=1,\dots ,n}$. The index ${\mathrm{ind}}_{a}f$ of a function f in a point a is the index of a bilinear form given by the matrix A. The Morse Lemma shows that the behavior of f at a can be completely described by this index (see [1]). What is more, ${\mathrm{ind}}_{a}f$ is invariant under diffeomorphisms (see Proposition 10).

Proposition 9.

Let f be a $C^{\infty }$-class function in an environment of a point $a\in {\mathbb{R}}^n$, $n⩾1$, such that there exists an integer number $l\in \{0,1,\dots ,n\}$, and

$$f\left(x\right)=-\sum _{i=1}^{\mathit{l}}{(x_i-a_i)}^2+\sum _{i=\mathit{l}+1}^n{(x_i-a_i)}^2,$$

in a certain environment of a. Then, $l={\mathrm{ind}}_{a}f$.

Proof. 

We have

$${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{x}_i}}\left(x\right)=\left\{\begin{array}{cc}-2(x_i-a_i)\hfill & \mathrm{for}i=1,\dots ,l,\hfill \\ 2(x_i-a_i)\hfill & \mathrm{for}i=l+1,\dots ,n.\hfill \end{array}\right.$$

So,

$${\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)=\left\{\begin{array}{cc}-2\hfill & \mathrm{for}i=j=1,\dots ,l,\hfill \\ 2\hfill & \mathrm{for}i=j=l+1,\dots ,n,\hfill \\ 0\hfill & \mathrm{for}i\ne j.\hfill \end{array}\right.$$

Let $A={\left[a_{i,j}\right]}_{i,j=1,\dots ,n}={\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]}_{i,j=1,\dots ,n}$ and $F:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ be a bilinear form given by a matrix A. Then, for $v,w\in {\mathbb{R}}^n$, we get

$$F(v,w)=\sum _{i,j=1}^n{a}_{i,j}{v}_i{w}_j=\sum _{j=1}^n{a}_{j,j}{v}_j{w}_j=-2\sum _{j=1}^{\mathit{l}}{v}_j{w}_j+2\sum _{j=\mathit{l}+1}^n{v}_j{w}_j.$$

So,

$$F(v,v)=-2\sum _{j=1}^{\mathit{l}}{v}_j^2+2\sum _{j=\mathit{l}+1}^n{v}_j^2.$$

Now let

$$R_1=\{v\in {\mathbb{R}}^n:v=(v_1,\dots ,v_l,0,\dots ,0),v_i\in {\mathbb{R}}^n,i=1,\dots ,l\},$$

$$R_2=\{v\in {\mathbb{R}}^n:v=(0,\dots ,0,v_{l+1},\dots ,v_n),v_i\in {\mathbb{R}}^n,i=l+1,\dots ,n\}.$$

Therefore, $R_1$, $R_2$, and $R_1+R_2$ are linear subspaces of ${\mathbb{R}}^n$, $R_1\cap R_2=\left\{0\right\}$ and for any $v\in {\mathbb{R}}^n$:

$$\begin{array}{cc}\hfill v& =(v_1,\dots ,v_l,v_{l+1},\dots ,v_n)\hfill \\ \hfill & =(v_1,\dots ,v_l,0,\dots ,0)+(0,\dots ,0,v_{l+1},\dots ,v_n)\in R_1+R_2.\hfill \end{array}$$

So, ${\mathbb{R}}^n=R_1\oplus R_2$, $\dim R_1=l$, $\dim R_2=n-l$, and

$$F(v,v)=\left\{\begin{array}{cc}-2\sum _{j=1}^{\mathit{l}}{v}_j^2<0\hfill & \mathrm{for}v\in R_1\setminus \left\{0\right\},\hfill \\ 2\sum _{j=\mathit{l}+1}^n{v}_j^2>0\hfill & \mathrm{for}v\in R_2\setminus \left\{0\right\}.\hfill \end{array}\right.$$

(32)

Let $V\subset {\mathbb{R}}^n$ be a linear subspace, such that $F(v,v)<0$ for all $v\in V\setminus \left\{0\right\}$. Then, for any $v\in V\cap R_2\setminus \left\{0\right\}$, by (32), we get $0<F(v,v)<0$. So, $V\cap R_2=\left\{0\right\}$. Therefore, by Proposition 1, we have

$$\dim V⩽\dim (V\cap R_2)+\dim {\mathbb{R}}^n-\dim R_2=0+n-(n-l)=l.$$

Finally, $l={\mathrm{ind}}_{a}f$. □

Proposition 10.

Let f be a $C^{\infty }$-class function in an environment of a point $a\in {\mathbb{R}}^n$, such that $f\left(a\right)=0$. If a is a non-degenerate critical point of a function f, and $\phi =({\phi }^1,\dots ,{\phi }^n):({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ is a diffeomorphism, then ${\mathrm{ind}}_{a}f=in{d}_a(f\circ \phi )$.

Proof. 

Let

$$A={\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]}_{i,j=1,\dots ,n},B={\left[{\displaystyle \frac{{\mathsf{\partial }}^2(f\circ \phi )}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]}_{i,j=1,\dots ,n},$$

and

$$C={\left[{\displaystyle \frac{\mathsf{\partial }{\phi }^j}{\mathsf{\partial }{x}_i}}\left(a\right)\right]}_{i,j=1,\dots ,n}.$$

Then, $detC\ne 0$, because $\phi$ is a diffeomorphism, and $B=C^{T}AC$. Now let F and G be two bilinear forms given by matrix A and matrix B, respectively. Let $V\subset {\mathbb{R}}^n$ be a linear subspace, such that $G(v,v)<0$ for all $v\in V\setminus \left\{0\right\}$ and $\dim V={\mathrm{ind}}_a(f\circ \phi )$. Let $L:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a linear mapping with matrix C. Then, for any $v\in V\setminus \left\{0\right\}$, we have

$$\begin{array}{cc}\hfill 0>G(v,v)=v^{T}Bv& =v^T{C}^{T}ACv={\left(Cv\right)}^{T}A\left(Cv\right)\hfill \\ \hfill & =F(Cv,Cv)=F\left(L\right(v),L(v\left)\right).\hfill \end{array}$$

Therefore, $L\left(V\right)\subset {\mathbb{R}}^n$ is a subspace, such that F is negatively defined on $L\left(V\right)$, and

$$\begin{array}{cc}\hfill \dim V& =\dim \left(\ker \left(L_{|V}\right)\right)+\dim \left(\mathrm{im}\left(L_{|V}\right)\right)=\dim \left(\mathrm{im}\left(L_{|V}\right)\right)\hfill \\ \hfill & =\dim \left(L\right(V\left)\right).\hfill \end{array}$$

Let $W\subset {\mathbb{R}}^n$ be a linear subspace, such that F is negatively defined on W. Because $detC\ne 0$, the matrix C is invertible, and

$${\left(C^{-1}\right)}^{T}B{C}^{-1}={\left(C^{-1}\right)}^T{C}^{T}AC{C}^{-1}=A.$$

Taking the linear mapping $L^{-1}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with the matrix $C^{-1}$, we get that

$$0>F(w,w)=G(L^{-1}\left(w\right),L^{-1}\left(w\right)),$$

for any $w\in W\setminus \left\{0\right\}$. So, G is negatively defined on the subspace $L^{-1}\left(W\right)\subset {\mathbb{R}}^n$. By the definition of V, we get

$$\begin{array}{cc}\hfill \dim W& =\dim \left(\ker \left(L_{|W}^{-1}\right)\right)+\dim \left(\mathrm{im}\left(L_{|W}^{-1}\right)\right)=\dim \left(\mathrm{im}\left(L_{|W}^{-1}\right)\right)\hfill \\ \hfill & =\dim \left(L^{-1}\left(W\right)\right)⩽\dim V=\dim \left(L\right(V\left)\right).\hfill \end{array}$$

Finally,

$${\mathrm{ind}}_{a}f=\dim \left(L\right(V\left)\right)=\dim V={\mathrm{ind}}_a(f\circ \phi ).$$

 □

Proposition 11.

Let f and g be two $C^{\infty }$-class functions in an environment of a point $a\in {\mathbb{R}}^n$, such that $f\left(a\right)=g\left(a\right)=0$. If f and g have non-degenerate critical points in a, then the following conditions are equivalent:

(a) 

There exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$, such that $f\circ \phi =g$ in an environment of a,

(b) 

${\mathrm{ind}}_{a}f={\mathrm{ind}}_{a}g$.

Proof. 

$\left(a\right)\Rightarrow \left(b\right)$ Let $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ be a diffeomorphism, such that $f\circ \phi =g$. Then, by Proposition 10, we get

$${\mathrm{ind}}_{a}g={\mathrm{ind}}_a(f\circ \phi )={\mathrm{ind}}_{a}f.$$

$\left(b\right)\Rightarrow \left(a\right)$ Suppose that ${\mathrm{ind}}_{a}f={\mathrm{ind}}_{a}g$. By the fact that f and g have non-degenerate critical points in a and using Theorem 10, we get that there exist diffeomorphisms $\overline{\phi }:({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$, $\overline{\psi }:({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer numbers $l,k\in \{0,1,\dots ,n\}$, such that

$$\left\{\begin{array}{cc}& f\circ \overline{\phi }=-\sum _{i=1}^{\mathit{l}}{(x_i-a_i)}^2+\sum _{i=\mathit{l}+1}^n{(x_i-a_i)}^2,\hfill \\ \hfill & g\circ \overline{\psi }=-\sum _{i=1}^{\mathit{k}}{(x_i-a_i)}^2+\sum _{i=\mathit{k}+1}^n{(x_i-a_i)}^2,\hfill \end{array}\right.$$

(33)

in a certain environment of a. By Propositions 9 and 10, we get

$$l=in{d}_a(f\circ \overline{\phi })=in{d}_{a}f=in{d}_{a}g=in{d}_a(g\circ \overline{\psi })=k.$$

So, using (33), we have

$$f\circ \overline{\phi }=g\circ \overline{\psi },$$

in a certain environment of a.

Taking $\phi =\overline{\phi }\circ {\overline{\psi }}^{-1}$, we get the assertion. □

6. Symmetric Tensors

Modern geometry has evolved recently. The issue of differential operators in the bundles of symmetric tensors on a Riemannian manifold is a subject of interest for mathematicians, physicists, and scholars of other branches of science. As a very important part of this area of investigation, symmetric tensors are a theme of interest, e.g., [25,26].

All the objects and morphisms are assumed to be smooth.

Let $(M,g)$ be an oriented Riemannian manifold of dimension n, where $M={\mathbb{R}}^n$, and g is the standard flat metric. Then, ${\mathsf{\partial }}_j={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}^j}}$ and $d{x}^j$, for $j\in \{1,\dots ,n\}$, form dual bases of the tangent bundle $T=TM$ and the cotangent bundle $T^{*}=T^{*}M$, respectively. For any vector bundle E over M, we denote by $C^{\infty }\left(E\right)$ the $C^{\infty }\left(M\right)$ - module of sections of E, where $C^{\infty }\left(M\right)$ is the ring of smooth functions on M.

Recall that a k-tensor $\phi$ is called symmetric if $\phi (X_{\sigma \left(1\right)},\dots ,X_{\sigma \left(k\right)})=\phi (X_1,\dots ,X_k)$ for all $X_1,\dots ,X_k\in T$ and any permutation $\sigma$ of the set $\{1,\dots ,k\}$.

For $\phi \in S^k$, $\psi \in S^l$,

$$\begin{array}{cc}\hfill & (\phi \odot \psi )(X_1,\dots ,X_k,X_{k+1},\dots ,X_{k+l})\hfill \\ \hfill & =\sum _{\sigma \in sh(k,l)}\phi (X_{\sigma \left(1\right)},\dots ,X_{\sigma \left(k\right)})\psi (X_{\sigma (k+1)},\dots ,X_{\sigma (k+l)}),\hfill \end{array}$$

where $X_1,\dots ,X_k,X_{k+1},\dots ,X_{k+l}\in T$, and where $sh(k,l)$ denotes the set of all shuffles of type $(k,l)$, i.e., the set of all permutations of the set $\{1,\dots ,k+l\}$, which are increasing on each of the two sets $\{1,\dots ,k\}$ and $\{k+1,\dots ,k+l\}$ (see also [27]).

Let ${\mathrm{d}}^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^{k+1}\right)$ be the operator of the symmetric derivative, where $S^k=S^{k}M$ is the bundle of k-symmetric tensors (k-forms).

Proposition 12.

Let $\phi \in C^{\infty }\left(S^k\right)$, $X_1,\dots ,X_{k+1}\in C^{\infty }\left(T\right)$. Then,

$${\mathrm{d}}^{\mathrm{s}}\phi (X_1,\dots ,X_{k+1})=\sum _{j=1}^{k+1}\left({\nabla }_{X_j}\phi \right)(X_1,\dots ,\widehat{X_j},\dots ,X_{k+1}),$$

(34)

where ∇ denotes the connection in $S^k$.

Proof. 

See [25]. □

Proposition 13

(See also [25]). Let f be a function on ${\mathbb{R}}^n$. Then, for all ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\cup \left\{0\right\}$, the following holds:

$${\mathrm{d}}^{\mathrm{s}}(f{d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})=\mathrm{d}f\odot {d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}.$$

(35)

Proof. 

Let $k\in \mathbb{N}\cup \left\{0\right\}$, and let ${\alpha }_1+\dots +{\alpha }_n=k$. By Proposition 12, we get

$$\begin{array}{cc}\hfill & {\mathrm{d}}^{\mathrm{s}}(f{d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})({\mathsf{\partial }}_{j_1},\dots ,{\mathsf{\partial }}_{j_{k+1}})\hfill \\ \hfill & =\sum _{l=1}^{k+1}{\nabla }_{{\mathsf{\partial }}_{j_l}}(f{d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})({\mathsf{\partial }}_{j_1},\dots ,\widehat{{\mathsf{\partial }}_{j_l}},\dots ,{\mathsf{\partial }}_{j_{k+1}})\hfill \\ \hfill & =\sum _{l=1}^{k+1}\left({\nabla }_{{\mathsf{\partial }}_{j_l}}f\right){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}({\mathsf{\partial }}_{j_1},\dots ,\widehat{{\mathsf{\partial }}_{j_l}},\dots ,{\mathsf{\partial }}_{j_{k+1}})\hfill \\ \hfill & +\sum _{l=1}^{k+1}f{\nabla }_{{\mathsf{\partial }}_{j_l}}({d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})({\mathsf{\partial }}_{j_1},\dots ,\widehat{{\mathsf{\partial }}_{j_l}},\dots ,{\mathsf{\partial }}_{j_{k+1}}).\hfill \end{array}$$

By the shape of frames, we can continue with

$$\begin{array}{cc}\hfill & =\sum _{l=1}^{k+1}\left({\nabla }_{{\mathsf{\partial }}_{j_l}}f\right){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}({\mathsf{\partial }}_{j_1},\dots ,\widehat{{\mathsf{\partial }}_{j_l}},\dots ,{\mathsf{\partial }}_{j_{k+1}})\hfill \\ \hfill & =\sum _{l=1}^{k+1}\left(\mathrm{d}f\right)\left({\mathsf{\partial }}_{j_l}\right){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}({\mathsf{\partial }}_{j_1},\dots ,\widehat{{\mathsf{\partial }}_{j_l}},\dots ,{\mathsf{\partial }}_{j_{k+1}})\hfill \\ \hfill & =\mathrm{d}f\odot {d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}({\mathsf{\partial }}_{j_1},\dots ,{\mathsf{\partial }}_{j_{k+1}}).\hfill \end{array}$$

 □

Corollary 6.

Let f be a function defined in a certain environment of the point $a\in {\mathbb{R}}^n$, $a=(a_1,\dots ,a_n)$, such that

$$f\left(a\right)=0,\mathrm{grad}f\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer number $0⩽\mathit{l}⩽n$, such that for all ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\cup \left\{0\right\}$, we have

$$\begin{array}{cc}\hfill & {\mathrm{d}}^{\mathrm{s}}((f\circ \phi ){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})\hfill \\ \hfill & =2\left(\sum _{j=l+1}^n(x_j-a_j)d{x}_j-\sum _{j=1}^l(x_j-a_j)d{x}_j\right)\odot {d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n},\hfill \end{array}$$

in a certain environment of the point a.

Proof. 

This is a direct consequence of Theorem 10 and Proposition 13. □

Recall that the gradient operator $\mathrm{grad}:C^{\infty }\left(S^k\right)\to C^{\infty }(S^k\otimes T)$ is defined by

$$\mathrm{grad}=\mathfrak{a}{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}\mathfrak{a},$$

(36)

where, for $k=1,2,\dots$, $\mathfrak{a}:C^{\infty }\left(S^k\right)\to C^{\infty }(S^{k-1}\otimes T)$ is the operator given by

$$\mathfrak{a}\varphi =\sum _{i=1}^n\iota {\mathsf{\partial }}_i\varphi \otimes {\mathsf{\partial }}_i$$

(37)

and for $k=0$,

$$\mathfrak{a}\varphi =0,$$

where $\varphi \in C^{\infty }\left(S^k\right)$.

Notice that the $\mathrm{grad}$ operator in the case $k=0$ coincides with the classical gradient on functions.

Proposition 14.

Let f be a function on ${\mathbb{R}}^n$. Then, for all ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\cup \left\{0\right\}$, the following holds:

$$\mathrm{grad}(f{d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})={d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}\otimes \mathrm{grad}f.$$

(38)

Proof. 

See [25]. □

Corollary 7.

Let f be a function defined in a certain environment of the point $a\in {\mathbb{R}}^n$, $a=(a_1,\dots ,a_n)$, such that

$$f\left(a\right)=0,\mathrm{grad}f\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer number $0⩽\mathit{l}⩽n$, such that for all ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\cup \left\{0\right\}$, we have

$$\begin{array}{cc}\hfill & \mathrm{grad}((f\circ \phi ){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n})\hfill \\ \hfill & =2{d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}\otimes (a_1-x_1,\dots ,a_l-x_l,x_{l+1}-a_{l+1},\dots ,x_n-a_n),\hfill \end{array}$$

in a certain environment of the point a.

Proof. 

This is a direct consequence of Theorem 10 and Proposition 14. □

The trace operator $\mathrm{tr}:C^{\infty }(S^k\otimes T)\to C^{\infty }\left(S^{k-1}\right)$ is defined here by the formula

$$\mathrm{tr}(\varphi \otimes X)={\iota }_{\mathrm{X}}\varphi ,\mathrm{for}k>0,$$

(39)

and

$$\mathrm{tr}(\varphi \otimes X)=0,\mathrm{for}k=0,$$

where $\varphi \otimes X\in C^{\infty }(S^k\otimes T)$.

Recall also that the divergence operator $\mathrm{div}:C^{\infty }(S^k\otimes T)\to C^{\infty }\left(S^k\right)$ is given by

$$\mathrm{div}=\mathrm{tr}{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}\mathrm{tr}.$$

(40)

Proposition 15.

Let f be a function on ${\mathbb{R}}^n$. Then, for all ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\cup \left\{0\right\}$ and $j\in \{1,\dots ,n\}$, we have

$$\mathrm{div}(f{d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}\otimes {\mathsf{\partial }}_j)={\mathsf{\partial }}_j\left(f\right){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}.$$

(41)

Proof. 

See [25]. □

Corollary 8.

Let f be a function defined in a certain environment of the point $a\in {\mathbb{R}}^n$, $a=(a_1,\dots ,a_n)$, such that

$$f\left(a\right)=0,\mathrm{grad}f\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer number $0⩽\mathit{l}⩽n$, such that for all ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\cup \left\{0\right\}$ and $j\in \{1,\dots ,n\}$, the following holds:

$$\begin{array}{cc}\hfill & \mathrm{div}((f\circ \phi ){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n}\otimes {\mathsf{\partial }}_j)\hfill \\ \hfill & =\left\{\begin{array}{c}-2(x_j-a_j){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n},j\in \{1,\dots ,l\},\hfill \\ 2(x_j-a_j){d{x}_1}^{{\alpha }_1}\odot \dots \odot {d{x}_n}^{{\alpha }_n},j\in \{l+1,\dots ,n\},\hfill \end{array}\right.\hfill \end{array}$$

in a certain environment of the point a.

Proof. 

This is a direct consequence of Theorem 10 and Proposition 15. □

Now let $k\in \mathbb{N}\cup \left\{0\right\}$, $i_1,\dots ,i_n\in \mathbb{N}\cup \left\{0\right\}$, $i_1+\dots +i_n=k$.

Proposition 16.

Let $\varphi =f_{i_1\dots i_n}d{x}^{i_1}\odot \dots \odot d{x}^{i_n}$, where $f_{i_1\dots i_n}$ is a function on ${\mathbb{R}}^n$. Then,

$$\mathrm{div}\mathrm{grad}\varphi =\sum _{i=1}^n{\displaystyle \frac{{\mathsf{\partial }}^2{f}_{i_1\dots i_n}}{{\left(\mathsf{\partial }{x}^i\right)}^2}}d{x}^{i_1}\odot \dots \odot d{x}^{i_n}.$$

(42)

Proof. 

See the proof of Theorem 5.35 in [25]. □

Proposition 17.

Let $f_{i_1\dots i_n}$ be a function defined in a certain environment of the point $a\in {\mathbb{R}}^n$, $a=(a_1,\dots ,a_n)$, such that

$$f_{i_1\dots i_n}\left(a\right)=0,\mathrm{grad}{f}_{i_1\dots i_n}\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^2{f}_{i_1\dots i_n}}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer number $0⩽\mathit{l}⩽n$, such that

$${\displaystyle \frac{{\mathsf{\partial }}^2(f_{i_1\dots i_n}\circ \phi )}{{\left(\mathsf{\partial }{x}^i\right)}^2}}=\left\{\begin{array}{c}-2,\mathrm{if}0<i⩽l\hfill \\ 2,\mathrm{if}l+1⩽i⩽n,\hfill \end{array}\right.$$

in a certain environment of the point a.

Proof. 

Using the Morse Lemma (Theorem 10) and taking the respectable second partial derivatives, we get the assertion. □

Proposition 18.

Let $\varphi =f_{i_1\dots i_n}d{x}^{i_1}\odot \dots \odot d{x}^{i_n}$, where $f_{i_1\dots i_n}$ is a function defined in a certain environment of the point $a\in {\mathbb{R}}^n$, $a=(a_1,\dots ,a_n)$, such that

$$f_{i_1\dots i_n}\left(a\right)=0,\mathrm{grad}{f}_{i_1\dots i_n}\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^2{f}_{i_1\dots i_n}}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer number $0⩽\mathit{l}⩽n$, such that

$$\mathrm{div}\mathrm{grad}(\varphi \circ \phi )=4\left({\displaystyle \frac{n}{2}}-l\right)d{x}^{i_1}\odot \dots \odot d{x}^{i_n},$$

(43)

in a certain environment of the point a.

Here, $\varphi \circ \phi =(f_{i_1\dots i_n}\circ \phi )d{x}^{i_1}\odot \dots \odot d{x}^{i_n}$.

Proof. 

This is a direct consequence of Proposition 16 and Proposition 17. □

Let $a\in {\mathbb{R}}^n$. We denote by $\mathfrak{F}$ the set of functions of the form:

$$\{f:({\mathbb{R}}^n,a)\to \mathbb{R}:f\left(a\right)=0,\mathrm{grad}f\left(a\right)=0,det\left[{\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0\},$$

and let $\mathfrak{D}=\{\varphi \in C^{\infty }\left(S^k\right):\varphi =f_{i_1\dots i_n}d{x}^{i_1}\odot \dots \odot d{x}^{i_n},f_{i_1\dots i_n}\in \mathfrak{F}\}$.

Consider the following equation:

$$\mathrm{div}\mathrm{grad}\varphi =0\mathrm{for}\varphi \in \mathfrak{D}.$$

(44)

By Proposition 18, we have the following:

Corollary 9.

Assume that $n\in 2\mathbb{N}$. For solving Equation (44), it is enough to find all $f_{i_1\dots i_n}\in \mathfrak{F}$ for which $l={\displaystyle \frac{n}{2}}$, i.e., if n is even, solutions exist when $l=n/2$.

Finally, the Sampson Laplacian ${\Delta }^s:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^k\right)$ is given by

$${\Delta }^s={{\mathrm{d}}^{\mathrm{s}}}^{*}{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{*},$$

(45)

where ${{\mathrm{d}}^{\mathrm{s}}}^{*}$ denotes the operator formally adjoint to ${\mathrm{d}}^{\mathrm{s}}$ with respect to the global scalar product (see, e.g., [25,26]).

Theorem 12.

$${\Delta }^s=-\mathrm{div}\mathrm{grad}.$$

(46)

Proof. 

See [25,26]. □

Proposition 19.

Let $\varphi =f_{i_1\dots i_n}d{x}^{i_1}\odot \dots \odot d{x}^{i_n}$, where $f_{i_1\dots i_n}$ is a function defined in a certain environment of the point $a\in {\mathbb{R}}^n$, $a=(a_1,\dots ,a_n)$, such that

$$f_{i_1\dots i_n}\left(a\right)=0,\mathrm{grad}{f}_{i_1\dots i_n}\left(a\right)=0,\mathit{and}det\left[{\displaystyle \frac{{\mathsf{\partial }}^2{f}_{i_1\dots i_n}}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}\left(a\right)\right]\ne 0.$$

Then, there exists a diffeomorphism $\phi :({\mathbb{R}}^n,a)\to ({\mathbb{R}}^n,a)$ and the integer number $0⩽\mathit{l}⩽n$, such that

$${\Delta }^s(\varphi \circ \phi )=-4\left({\displaystyle \frac{n}{2}}-l\right)d{x}^{i_1}\odot \dots \odot d{x}^{i_n},$$

(47)

in a certain environment of the point a.

Proof. 

This is a direct consequence of Theorem 12 and Proposition 18. □

Consider now the following equation:

$${\Delta }^s\varphi =0\mathrm{for}\varphi \in \mathfrak{D}.$$

(48)

By Proposition 19, we obtain the following:

Corollary 10.

Assume that $n\in 2\mathbb{N}$. For solving Equation (48), it is enough to find all $f_{i_1\dots i_n}\in \mathfrak{F}$ for which $l={\displaystyle \frac{n}{2}}$, i.e., if n is even, solutions exist when $l=n/2$.

Some differential equations are strictly related to the Morse Theory. For example, the behavior of finite Morse index solutions of the equation

$$-\Delta u={\left|x\right|}^{\alpha }{\left|u\right|}^{p-1}u,u\in \Omega \subset {\mathbb{R}}^N,$$

where $p>1$, $\alpha >-2$, and $\Omega$ is a bounded (respectively, unbounded) domain, has been studied in the context of stable solutions (see [4]). More precisely, the Morse Lemma in the context of some differential equations still has rich potential, e.g., for degenerate critical points of a function u, which satisfies the following:

$$-\Delta u=f\left(u\right)\mathrm{in}{B}_1,$$

(49)

where $u\in C^2\left(B_1\right)$, $B_1$ is the unit ball of ${\mathbb{R}}^2$, and f is a smooth nonlinearity. The Morse Lemma, for solutions to such equations as (49), supplies a precise qualitative information on the shape of u in a neighborhood of the critical points. This information is local but jointly with some global properties such as Dirichlet boundary conditions enable deriving important properties of u like the number of critical points or the shape of the level sets (see [13]).

There are some works on approximation theory and operators that can contextualize this section’s applications. For example, the paper [14] introduces the development of wavelet-enhanced King-type Baskakov operators preserving quadratic test functions. In that paper, the explicit formulas for the moments of these operators are given, and approximation results by Korovkin-type theorems are demonstrated. The rate of convergence is measured there, using many important notions, such as the modulus of continuity, Lipschitz spaces, and statistical approximation, illustrating improved convergence characteristics. It is worth noting that, in paper [14], quantitative error boundaries are established by direct approximation theorems, with numerical validation that demonstrates a 20–50 per cent reduction in error for oscillatory functions. What is more, Baskakov operators preserving quadratic forms align with Morse normal forms.

Another important paper [15] is associated with a study of general Appell polynomials: a new sequence of Szász-Integral type of sequence of operators via general Appell polynomials is introduced there to discuss approximation properties for Lebesgue integrable functions. Moreover, estimates in view of the test functions and central moments are studied. The paper [15] gives the convergence rate discussion using the Korovkin and Voronovskaja-type theorems, weighted approximation results, and statistical approximation theorems. In our case, the most valuable application is the fact that general Appell polynomials can extend approximation tools for divergence or Laplacian operators.

Another interesting paper is [16] that illustrates the approximation and related properties of Szász–Jakimovski–Leviatan-type operators constructed using Beta functions, which are introduced through the Appell polynomials in Dunkl formulations. There, the approximation is studied in Korovkin’s and weighted Korovkin’s spaces involving local and global approximations. As a consequence, approximations are studied in [16] through A-statistical convergence. For our consideration, Beta-type integral operators are potentially useful to improve function approximations in $C^{\infty }$ settings.

The most important applications of our investigations could be found in so called Floer theory. Floer homology as an infinite-dimensional analogue of Morse homology, plays a crucial role in symplectic geometry, especially in the proof of the Arnold conjecture. This problem is of classical mechanics origin and concerns the number of 1-periodic trajectories of a non-degenerate Hamiltonian system (see [12]).

Morse theory as a fundamental tool of differential topology also delivers important notions like Morse inequalities, which give a lower bound for the number of critical points of a function (see [12]). Additionally, in the article [11], Morse theory has been considered for perturbed Dirac-harmonic maps into flat tori. There, Morse homology is defined for some classes of perturbations and determined by a homotopy type of the perturbations. The research in [11] might be able to provide some other directions for symmetric tensors considerations in the practical applications of such objects with a Riemannian metric g as spin manifolds. Notably, in physics, models with various types of perturbations are very precious (see also, e.g., [11]).