Rigidity of Holomorphically Projective Mappings of Kähler and Hyperbolic Kähler Spaces with Finite Complete Geodesics

Abstract

In the paper, we consider holomorphically projective mappings of n-dimensional pseudo-Riemannian Kähler and hyperbolic Kähler spaces. We refined the fundamental linear equations of the above problems for metrics of differentiability class $C^2$. We have found the conditions for n complete geodesics and their image that must be satisfied for the holomorphically projective mappings to be trivial, i.e., these spaces are rigid with precision to affine mappings.

1. Introduction

This paper develops new ideas in the theory of holomorphically projective mappings of (classical) Kähler and hyperbolic Kähler spaces. These questions are related to compact and complete geodesics, Kähler spaces, and their holomorphically projective mappings and transformations.

In 1954, the study of geodesic mappings of (classical) Kähler spaces began [1,2,3]. This question is closely connected with equidistant spaces [4,5,6,7,8,9]. Equidistant spaces, as established by Sinyukov [10,11], admit convergent vector fields (shown by Shirokov [12,13,14]), which are special concircular vector fields by Yano [15]. These vector fields probably first appeared during Brinkman’s study of conformal mappings between Einstein spaces [16] (see Petrov’s monographs [17,18]). Possible applications in theoretical physics are listed here. At present, in this direction, we can recommend Hall’s monograph [19].

In 1954 [20], the study of the holomorphically projective mappings of classical Kähler spaces also began, which are generalizations of geodesic mappings. An overview of the work up until 1963 can be found, for example, in [21,22,23].

Mikeš extended these results in various directions [4,24]. Some of these extensions are discussed in the fifth and final chapter of Sinyukov’s monograph [11]. Further details can be found in Mikeš’ dissertation [4], especially concerning $K_n\left[B\right]$ (see Section 4). These and other results are described in more detail in [6,7,8,9].

Other problems and ideas in the theory of holomorphically projective mappings were advanced by Aminova and Kalinin [25,26,27,28] and others. Among other things, these authors found all metrics of four-dimensional (pseudo-) Riemannian (classical) Kähler spaces that admit holomorphically projective mappings.

The complex projective space $(\mathbb{CP}\left(n\right),g_{\mathrm{Fubini}-\mathrm{Study}})$ admits global, non-trivial holomorphically projective mappings and transformations with maximal parameters, as shown in [29,30]. Locally, the same was proven for spaces with constant holomorphic curvature in [24]. These calculations are primarily carried out in complex forms, as in [30].

Holomorphically projective mappings of hyperbolic and parabolic Kähler spaces have also been studied by Prvanović [31], Kurbatova [32,33], and Shiha et al. [34]. Holomorphically projective mappings have been generalized in many ways [35].

In 1962, Petrov [36] studied quasi-geodesic mappings and showed their applications in simulating physical processes and electromagnetic fields. Similar results are presented by Bejan and Kowalski [37]. These mappings generalize F-planar mappings by Mikeš and Sinyukov. This result was supplemented in [38]. More results about F-planar mappings and transformations are detailed in [7,8,9].

Almost-geodesic mappings of type ${\pi }_2$ are also a direct generalization of holomorphically projective mappings [7,8,9,11,39]. In a 2019 paper by Kozak and Borowiec [40], almost-geodesic mappings were given a new physical interpretation as transformations that genuinely preserve geodesics in spacetime. These topics have been addressed in numerous monographs and reviews, such as [19,41,42,43,44,45,46].

Many authors have explored rigidity problems, i.e., conditions under which holomorphically projective mappings are affine (trivial). We build upon these studies of rigidity, focusing on similar problems for motions (Killing vector fields) and their generalizations in compact or complete Riemannian and Kähler spaces, as discussed in Yano and Bochner’s monographs [22,23]. Using Bochner’s methods, further advancements were made by Stepanov [47], Tachibana and Ishihara [48,49], Hasegawa and Yamauchi [50], and Akbar-Zadeh and Couty [51,52,53]. Later, Sinyukov, Sinyukova [54] and Mikeš [7] expanded on this research, and more general results were derived using methods by Švec [55], as shown in [7,8,9].

In 1961, Tachibana and Ishihara [56] proved that Ricci-symmetric (non-Einstein) spaces do not admit non-trivial analytic holomorphically projective transformations. In 1979, Mikeš [4,7] extended this result by showing that these spaces also do not admit non-trivial mappings, even without assuming global requirements. Additional contributions were made by Bácsó and Ilosvay [57].

Sakaguchi [58], using Sinyukov’s methods [10], proved that symmetric and recurrent Kähler spaces with non-constant holomorphically projective curvature do not admit non-trivial holomorphically projective mappings. These results were further generalized by Domashev and Mikeš [24] for (pseudo-) Kähler spaces.

Later, many of the above results were proved by Kurbatova [32,33] for holomorphically projective mappings of hyperbolically Kähler spaces. Analogically, results for parabolic Kähler spaces were obtained by Shiha [34].

We specify the fundamental equations of holomorphically projective mappings of Kähler and hyperbolically Kähler spaces, Theorem 1. Based on Lemma 3, old locally results about rigidity are precised. The main results of our study are Theorems 2 and 3. They demonstrate that for a mapping to be rigid, the space does not need to be complete. It is sufficient for a finite number of geodesics and their images to be complete. In other words, the space is uniquely defined by specific geodesics, which form the supporting skeleton (reinforcement) of the space. With these results, we supplement and correct our article [35].

2. Kähler and Hyperbolic Kähler Spaces

This section introduces the fundamental concepts of the theory of Kähler and hyperbolic Kähler manifolds (or spaces), which will be useful in the subsequent discussion. These manifolds can be defined in several equivalent ways. A Kähler or hyperbolic Kähler manifold carries three compatible structures:

$$K_n=(M_n,g,F),$$

where $M_n$ is a manifold, g is a metric, and F$(F\ne \alpha Id)$ is a structure tensor of $K_n$, which satisfies the relations

$$F^2=eId,g(X,FX)=0\mathrm{and}\nabla F=0,$$

where $e=\mp 1$, ∇ is Levi-Civita connection, and X is any tangent vector on $K_n$. Necessarily, the spaces $K_n$ are of an even dimension, i.e., $n=2m$, and $n\ge 4$. We assume that the metric g has an arbitrary admissible signature.

For $e=-1$, we have a Kähler space $K_n^{-}$, and for $e=+1$ – hyperbolic Kähler space $K_n^{+}$. The structure F is called complex and a product structure, respectively.

In local coordinates $x\equiv (x^1,x^2,\dots ,x^n)$, components $g_{ij}\left(x\right)$ and $F_i^h\left(x\right)$ of g and F satisfy the relations

$$F_{\alpha }^h{F}_i^{\alpha }=e{\delta }_i^h;F_{(i}^{\alpha }{g}_{j)\alpha }=0;F_{i,j}^h=0.$$

Here and in what follows, “,” denotes a covariant derivative on $K_n$ and the round brackets denote symmetrization of indices without division.

The spaces $K_n^{-}$ were first considered by Shirokov [13]. Independently, in a complex form, these spaces were studied by Kähler [59]. In the available literature, these spaces are also called Kählerian. We will present the notation that is used in Mikeš dissertations [4], and in many other articles, such as [7,8,9,11,21,22]. In the same way, the results about holomorphically projective mappings of hyperbolic Kähler spaces $K_n^{+}$ were studied by Prvanović [31] and Kurbatova [33]. Detailed results can be found in our monograph [9].

In the Kähler spaces $K_n$, we shall introduce the operation of index conjugation as follows:

$$A_{\cdots \overline{i}\cdots }^{\cdots }\equiv A_{\cdots \alpha \cdots }^{\cdots }{F}_i^{\alpha };A_{\cdots }^{\cdots \overline{j}\cdots }\equiv A_{\cdots }^{\cdots \alpha \cdots }{F}_{\alpha }^j.$$

According to the definition of a tensor F, this operation has the following properties:

$$A_{\overline{\overline{i}}}=e{A}_i;B^{\overline{\overline{i}}}=e{B}^i;A_{\overline{\alpha }}{B}^{\alpha }=A_{\alpha }{B}^{\overline{\alpha }};A_{\overline{\alpha }}{B}^{\overline{\alpha }}=e{A}_{\alpha }{B}^{\alpha };{\left(A_{\overline{i}}\right)}_{,j}=A_{\overline{i},j};{\left(B^{\overline{i}}\right)}_{,j}=B_{,j}^{\overline{i}}.$$

For the Kronecker symbol, metric and its inverse tensors, the following holds:

$${\delta }_i^{\overline{h}}={\delta }_{\overline{i}}^h=F_i^h;g_{\overline{i}j}+g_{i\overline{j}}=0;g_{\overline{i}\overline{j}}=-e{g}_{ij};g^{\overline{i}j}+g^{i\overline{j}}=0;g^{\overline{i}\overline{j}}=-e{g}^{ij}.$$

For the Riemann and Ricci tensors, we have $R_{ijk}^h={\partial }_j{\mathrm{\Gamma }}_{ik}^h-{\partial }_k{\mathrm{\Gamma }}_{ij}^h+{\mathrm{\Gamma }}_{\alpha j}^h{\mathrm{\Gamma }}_{ik}^{\alpha }-{\mathrm{\Gamma }}_{\alpha k}^h{\mathrm{\Gamma }}_{ij}^{\alpha }$, ${\partial }_j=\partial /\partial x^j$, and $R_{ik}=R_{i\alpha k}^{\alpha }$; moreover, the following formulas hold:

$$R_{hi\overline{j}\overline{k}}=-e{R}_{hijk}\equiv g_{h\alpha }{R}_{ijk}^{\alpha };R_{\overline{i}\overline{j}}=-e{R}_{ij};R_{\alpha jk}^{\overline{\alpha }}=2R_{\overline{j}k}.$$

In the Kähler spaces $K_n$ we can consider the holomorphically projective curvature tensor

$$P_{ijk}^h\equiv R_{ijk}^h+{\displaystyle \frac{1}{n+2}}({\delta }_k^h{R}_{ji}-{\delta }_j^h{R}_{ki}-e{\delta }_{\overline{k}}^h{R}_{\overline{j}i}+e{\delta }_{\overline{j}}^h{R}_{\overline{k}i}+2e{\delta }_{\overline{ı}}^h{R}_{\overline{j}k}).$$

When certain maps f of spaces are considered, e.g., $K_n\stackrel{f}{\to }{\overline{K}}_n$, both spaces are assigned to the own coordinate system x, which are general with respect to these mappings. In this coordinate system, the corresponding points $x\in K_n$ and $f\left(x\right)\in {\overline{K}}_n$ have the same coordinates $x\equiv \left(x^1,x^2,\dots ,x^n\right)$.

In this case, we denote the corresponding geometric objects in ${\overline{K}}_n$ with a bar. For instance, ${\overline{R}}_{ijk}^h$ and ${\overline{R}}_{ij}$ are the Riemannian and Ricci tensors.

3. General Questions Concerning Holomorphically Projective Mappings of Kähler and Hyperbolic Kähler Spaces

Natural generalizations of geodesic mappings are the holomorphically projective mappings (HP-mappings) of $K_n^{-}$ spaces and hyperbolic spaces $K_n^{+}$. Similar problems arise in the HP-mapping theory, much like in the geodesic mapping theory. Interestingly, many results that hold for geodesic mappings extend naturally to HP-mappings, indicating a high degree of compatibility between the two.

Note that HP-mappings are generally considered under the condition of structure preservation. It has been shown that, in the case of HP-mappings, the structure is necessarily preserved.

The works by Otsuki and Tashiro [20,60], Ishihara [48], Domashev and Mikeš [24], and those by Mikeš [7,8,9] are devoted to general questions concerning the theory of HP-mappings of $K_n^{-}$ spaces.

Problems related to integrating the fundamental equations of the HP-mapping theory, along with other related questions, are being examined in the works of Aminova and Kalinin [25,26,27,28].

The fundamentals of the theory of HP-mappings of $K_n^{-}$ spaces can be found in the works of Beklemishev [21], Yano [22,23], Sinyukov [11], and Mikeš [7,8,9]. In [11] (5th chapter), Sinyukov presented classical results on HP-mappings, while further results were obtained by Mikeš and Domashev [24], and by Mikeš [4] (see [7,8,9]). Similarly, results for hyperbolic $K_n^{+}$ spaces were obtained by Kurbatova [33].

Below, the terms regarding holomorphically projective mappings and transformations are presented in detail. For instance, this applies to (classical) Kähler spaces $K_n^{-}$ in [11,20,21,22] and hyperbolic Kähler spaces $K_n^{+}$ in [32,33]. Here, we outline the fundamental definitions, applicable to both classes of spaces $K_n^{-}$ and $K_n^{+}$, in a manner analogous to those found in publications [7,8,9].

3.1. Definitions and the Basic Equations of Analytically Planar Curves on Kähler Spaces

Definition 1.

An analytically planar curve γ of the Kähler space $K_n$ is a curve defined by the equations $\gamma =\gamma \left(t\right)$ whose tangent vector $\lambda =d\gamma \left(t\right)/dt$, being translated, remains in the area element formed by the tangent vector λ and its conjugate $F\lambda$.

It means that the analytically planar curve $\gamma$ is characterized by the conditions

$${\nabla }_t\lambda ={\rho }_1\left(t\right)\lambda +{\rho }_2\left(t\right)F\lambda ,$$

where ${\rho }_1$, ${\rho }_2$ are functions of the argument t, and ${\nabla }_t$ is covariant derivative along $\gamma$.

The physical meaning of these curves is outlined in [61,62].

If ${\rho }_2\left(t\right)\equiv 0$, then $\gamma$ is a geodesic; moreover, if ${\nabla }_{\tau }\lambda =0$ holds, then the parameter $\tau$ on a geodesic $\gamma \left(\tau \right)$ is called canonical (or affine). In this case, the length of the tangent vector $\lambda \left(\tau \right)$ has a constant length, including zero, i.e., $g\left(\lambda \right(\tau ),\lambda (\tau \left)\right)=$ const. Parameter s is lenght if $g\left(\lambda \right(s),\lambda (s\left)\right)=\epsilon =\pm 1$. In the case $g\left(\lambda \right(s),\lambda (s\left)\right)=0$, $\gamma$ is called an isotropic geodesic. Evidently, parameters $\tau$ and s are bounded by the following linear relation $\tau =\alpha s+\beta$, where $\alpha$ and $\beta$ are constants.

Analogically, the parameter $\tau$ of the analytically planar curve $\gamma \left(\tau \right)$ is called canonical if the tangent vector $\lambda \left(\tau \right)$ satifies ${\nabla }_{\tau }\lambda ={\rho }_2\left(\tau \right)F\lambda$. For canonical parameter $\tau$, the tangent vector $\lambda \left(\tau \right)$ has a constant length, $g\left(\lambda \right(\tau ),\lambda (\tau \left)\right)=$ const. In some cases, we will include curves with isotropic directions, i.e., $g\left(\lambda \right(\tau ),\lambda (\tau \left)\right)=0$.

We note that it is always possible to transition to the canonical parameter $\tau$, i.e., to find a parameter $\tau$ such that the function ${\varrho }_1$ vanishes. Furthermore, it follows from ODEs’ theory that, for a given continuous function ${\varrho }_2\left(\tau \right)$, there is a unique analytically planar curve $\gamma \left(\tau \right)$ passing through a given point and direction.

Moreover, these curves have similar properties to geodesics:

Lemma 1.

If the tangent vector λ of an analytically planar curve γ is isotropic (null-vector) in one of its points, then it is isotropic in all its points γ.

3.2. Definitions and the Basic Equations of Holomorphically Projective Mappings

Definition 2.

The diffeomorphism of $K_n$ onto ${\overline{K}}_n$ is a holomorphically projective mapping, if it transforms all analytically planar curves of $K_n$ onto analytically planar curves of ${\overline{K}}_n$.

Under the HP-mapping, the structure of the spaces $K_n$ and ${\overline{K}}_n$ is preserved. That is, in the coordinate system x, general with respect to the mapping, the conditions ${\overline{F}}_i^h\left(x\right)=F_i^h\left(x\right)$ are satisfied. To be more precise, ${\overline{F}}_i^h\left(x\right)=\pm F_i^h\left(x\right)$ for $K_n$, since the structure in $K_n$ is defined with an accuracy to be within the sign (see [9], p. 160).

The holomorphically projective mappings were introduced by Otsuki and Tashiro [20] for $K_n^{-}$, and Prvanović [31] for $K_n^{+}$ under the a priori assumption that the structure was preserved.

Note that HP-mappings are special F-planar mappings introduced by Mikeš and Sinyukov. Questions regarding the preservation of the structure for these mappings were studied in detail in [38].

It is well known [9] that the necessary and sufficient conditions for the holomorphically projective mappings of $K_n$ onto ${\overline{K}}_n$ encompass the fulfillment of the following conditions in the general (with respect to the mapping) coordinate system:

$${\overline{\mathrm{\Gamma }}}_{ij}^h\left(x\right)={\mathrm{\Gamma }}_{ij}^h\left(x\right)+{\psi }_i{\delta }_j^h+{\psi }_j{\delta }_i^h+e{\psi }_{\overline{i}}{\delta }_{\overline{j}}^h+e{\psi }_{\overline{j}}{\delta }_{\overline{i}}^h,$$

(1)

where ${\psi }_i$ is a vector, and ${\mathrm{\Gamma }}_{ij}^h$ and ${\overline{\mathrm{\Gamma }}}_{ij}^h$ are the Christoffel symbols of $K_n$ and ${\overline{K}}_n$. Relation (1) is equivalent to the following equations:

$${\overline{g}}_{ij,k}=2{\psi }_k{\overline{g}}_{ij}+{\psi }_i{\overline{g}}_{jk}+{\psi }_j{\overline{g}}_{ik}-e{\psi }_{\overline{i}}{\overline{g}}_{\overline{j}k}-e{\psi }_{\overline{j}}{\overline{g}}_{\overline{i}k}.$$

(2)

where ${\overline{g}}_{ij}$ are components of metric $\overline{g}$ on ${\overline{K}}_n$. When ${\psi }_i\not\equiv 0$, we say that the holomorphically projective mapping is non-trivial or affine. After contracting (1), it is valid that ${\psi }_i$ is necessarily a gradient; moreover,

$${\psi }_i={\partial }_i\mathrm{\Psi },\mathrm{where}\mathrm{\Psi }={\displaystyle \frac{1}{n+2}}ln\sqrt{\left|{\displaystyle \frac{det\overline{g}}{detg}}\right|}.$$

The Riemannian and Ricci tensors $K_n$ and ${\overline{K}}_n$ are connected by the following conditions:

$${\overline{R}}_{ijk}^h=R_{ijk}^h+{\delta }_k^h{\psi }_{ij}-{\delta }_j^h{\psi }_{ik}-e{\delta }_{\overline{k}}^h{\psi }_{i\overline{j}}+e{\delta }_{\overline{j}}^h{\psi }_{i\overline{k}}-2e{\delta }_{\overline{i}}^h{\psi }_{\overline{j}k};{\overline{R}}_{ij}=R_{ij}-(n+2){\psi }_{ij},$$

where ${\psi }_{ij}\equiv {\psi }_{i,j}-{\psi }_i{\psi }_j-e{\psi }_{\overline{i}}{\psi }_{\overline{j}}$ is a symmetric tensor, for which ${\psi }_{ij}=-e{\psi }_{\overline{i}\overline{j}}$.

The tensor $P_{ijk}^h$ is invariant of holomorphically projective mapping. It is vanishing if and only if $K_n$ is a space of constant holomorphic curvature. These spaces locally always admit holomorphically projective mapping onto a flat space [48,60]. Between two n-dimensional Kähler (resp. hyperbolically Kähler) spaces, non-trivial holomorphically projective mapping can be established (see [9], p. 486).

Mikeš [4] and Kurbatova [33] have found that the Kähler space $K_n$ admits a holomorphically projective mapping if and only if the system of equations

$$\begin{array}{c}\left(\mathrm{a}\right)a_{ij,k}={\lambda }_i{g}_{jk}+{\lambda }_j{g}_{ik}-e{\lambda }_{\overline{i}}{g}_{\overline{j}k}-e{\lambda }_{\overline{j}}{g}_{\overline{i}k};\hfill \\ \left(\mathrm{b}\right)n{\lambda }_{i,j}=\mu g_{ij}-a_{i\alpha }{R}_j^{\alpha }-a_{\alpha \beta }R_{·}^{\alpha }{}_{ij}{}_{·}{}^{\beta };\hfill \\ \left(\mathrm{c}\right){\mu }_{,i}=-2{\lambda }_{\alpha }{R}_i^{\alpha }\hfill \end{array}$$

(3)

has a solution for the unknown tensors $a_{ij}$, ${\lambda }_i$ and $\mu$, with $a_{ij}=a_{ji}=-e{a}_{\overline{i}\overline{j}},\left|a_{ij}\right|\ne 0$. The solutions of (2) and (3) are connected by the relations

$$a_{ij}=e^{2\psi }{\overline{g}}^{\alpha \beta }{g}_{\alpha i}{g}_{\beta j}\mathrm{and}{\lambda }_i=-e^{2\psi }{\overline{g}}^{\alpha \beta }{g}_{\alpha i}{\psi }_{\beta }.$$

Mapping is trivial if ${\lambda }_i=0$. Condition (3)(a) is necessary and sufficient for the existence of holomorphically projective mapping $K_n$; for $K_n^{-}$, it was obtained by Domashev and Mikeš [24].

Equation (3) forms a linear system of the Cauchy type with respect to the components of the unknown tensors $a_{ij}$, ${\lambda }_i$ and $\mu$. Consequently, the general solution of this system depends on $r_{hpm}\le {(n/2+1)}^2$ parameters [24]. For $r_{hpm}>2$, Equations (4) and (5) hold, (see [7,8,9,30]).

It is known [9], pp. 130–135, that for a sufficiently differentiable metric of $K_n$, the solution of Equation (3) in $K_n$ is reduced to the study of integrability conditions for (3) and its differential continuations, which, in turn, constitutes a system of linear algebraic equations for the unknowns $a_{ij}$, ${\lambda }_i$ and $\mu$. Thus, we can determine whether the given space $K_n$ admits holomorphically projective mapping, and if it does, to what degree of arbitrariness.

3.3. Holomorphically Projective Mappings and Killing Vector Fields

Evidently, because ${\lambda }_i={\partial }_i\left(2a_{\alpha \beta }{g}^{\alpha \beta }\right)$, covector ${\lambda }_i$ is the gradient, and, naturally, ${\lambda }_{i,j}={\lambda }_{j,i}$. In addition, for vector ${\lambda }_i$, it is valid that ${\lambda }_{\overline{i},\overline{j}}=-e{\lambda }_{i,j}$. Therefore, ${\lambda }_{\overline{i},j}+{\lambda }_{\overline{j},i}=0$. Consequently, the vector ${\lambda }_{\overline{i}}$ is a Killing vector.

The following lemma holds for Killing vector fields:

Lemma 2.

If the Killing vector field ξ is vanishing around some point, then it is vanishing everywhere.

Proof. 

Let $\xi$ be a Killing vector field on pseudo-Riemannian spaces. The associated covector field ${\xi }_i$ satisfies the Killing equation ${\xi }_{i,j}+{\xi }_{j,i}=0$. As is well known, this leads to the following formulas (see [9], p. 228):

$${\xi }_{i,j}={\xi }_{ij},{\xi }_{ij,k}={\xi }_{\alpha }{R}_{kji}^{\alpha }\mathrm{and}{\xi }_{ij}+{\xi }_{ji}=0.$$

These equations form a mixed system of Cauchy-type equations in the unknown functions ${\xi }_i\left(x\right)$ and ${\xi }_{ij}\left(x\right)$, which have a unique solution given the Cauchy initial conditions at the point $x_0$: ${\xi }_i\left(x_0\right)={\xi }_i^0$ and ${\xi }_{ij}\left(x_0\right)={\xi }_{ij}^0$.

Thus, if the vector field $\xi$ vanishes around some point $x_0$, then ${\xi }_i\left(x_0\right)=0$ and ${\xi }_{ij}\left(x_0\right)=0$. The solution satisfying these initial conditions is uniquely trivial, i.e., $\xi =0$. □

Since the vector field ${\lambda }_{\overline{i}}$ is Killing, the following lemma holds:

Lemma 3.

Let $K_n$ admit holomorphically projective mapping onto ${\overline{K}}_n$. Around some point, if this mapping is trivial (${\lambda }_i=0$), then it is trivial everywhere.

In many works [9,24,33,58], results on the triviality (or rigidity) of holomorphically projective mappings in special Kähler and hyperbolically Kähler spaces were proved. However, these results were only local. The above lemma extends them to the global case.

We note that Lemmas 2 and 3 hold in spaces where the manifold M is connected and path-connected, and may have boundary $\partial M$, where it must meet natural requirements. The differentiability class of the metric g is $C^2$, i.e., in a coordinate system x, the components of g are twice continuously differentiable, $g_{ij}\left(x\right)\in C^2$. In this case, the components $R_{ijk}^h\left(x\right)$ of the Riemann tensor are continuous functions.

Remark 1.

Therefore, we assume these conditions on the manifold M and the differentiability of $g\in C^2$ throughout this article, unless otherwise stated. Next, we suppose that there is a path-connected domain V in $M_n$ such that its boundary $\partial V$ is a Lipschitz boundary (see [63], p. 46). This condition is important for the possibility of solving the differential equations that occur in this work.

Recall that Formula (3)(c) was obtained under the condition $g,\overline{g}\in C^3$. We will show that from the above facts, this formula is obtained under the conditions $g,\overline{g}\in C^2$. As it was proven, the vector ${\lambda }_{\overline{i}}$ is Killing. That is why it is valid ${\lambda }_{\overline{i},jk}={\lambda }_{\overline{\alpha }}{R}_{kji}^{\alpha }$. From here, ${\lambda }_{i,jk}=e{\lambda }_{\overline{\alpha }}{R}_{kj\overline{i}}^{\alpha }$, and after contraction with $g^{ij}$, we obtain ${\left({\lambda }_{i,j}{g}^{ij}\right)}_{,k}=-2{\lambda }_{\alpha }{R}_k^{\alpha }$. Due to $\mu ={\lambda }_{i,j}{g}^{ij}$, we obtained Formula (3)(c). From what has been said, the following theorem follows:

Theorem 1.

The Kähler space $K_n$$\in C^2$ admits a holomorphically projective mapping onto ${\overline{K}}_n$ $\in C^2$ if and only if the system of Equation (3) has a solution for the unknown tensors $a_{ij}$, ${\lambda }_i$ and μ with $a_{ij}=a_{ji}=-e{a}_{\overline{i}\overline{j}},\left|a_{ij}\right|\ne 0$.

4. Holomorphically Projective Mappings of the Spaces ${\mathit{K}}_{\mathit{n}}\left[\mathit{B}\right]$

We denote the Kähler space $K_n$ by $K_n\left[B\right]$ if it admits a holomorphically projective mapping under which the relations (in detail, this is in Mikeš’ dissertation [4] (see [7,8]))

$$\left(\mathrm{a}\right)a_{ij,k}={\lambda }_i{g}_{jk}+{\lambda }_j{g}_{ik}-e{\lambda }_{\overline{i}}{g}_{\overline{j}k}-e{\lambda }_{\overline{j}}{g}_{\overline{i}k};\left(\mathrm{b}\right){\lambda }_{i,j}=\mu g_{ij}+B{a}_{ij}$$

(4)

are satisfied, where $a_{ij}$, $(=a_{ji}=a_{\overline{i}\overline{j}},|a_{ij}|\ne 0)$, ${\lambda }_i(\neg \equiv 0)$, $\mu$, B are tensors, while B is uniquely determined by the space $K_n$. When B is a constant, then ${\mu }_{,i}=2B{\lambda }_i$. When $B\equiv 0$, then $\mu$ is a constant. Relation (4) is equivalent to Relation (2) and

$${\psi }_{ij}=\overline{B}{\overline{g}}_{ij}-B{g}_{ij}.$$

(5)

As proven in [4], any holomorphically projective mappings on $K_n^{-}\left[B\right]$ must satisfy Equation (4) with the same B. For hyperbolically $K_n^{+}\left[B\right]$, this result can be demonstrated analogously. From Equation (5), it follows that $K_n\left[B\right]$ maps to ${\overline{K}}_n\left[\overline{B}\right]$. Furthermore, it can be shown that B and $\overline{B}$ are simultaneously constant.

These conditions are satisfied, in particular, under the holomorphically projective mappings between Einstein spaces, where $B=-{\displaystyle \frac{R}{n(n+2)}}$ and $\overline{B}=-{\displaystyle \frac{\overline{R}}{n(n+2)}}$, with R and $\overline{R}$ denoting the scalar curvatures of $K_n$ and ${\overline{K}}_n$, respectively.

Additionally, these results hold for spaces of constant holomorphic curvature, as discussed in [9], pp. 486–487.

Spaces in which K-concircular fields exist are $K_n\left[B\right]$ spaces. Equidistant vector fields necessarily exist in $K_n\left[0\right]$ spaces. The spaces $K_n\left[B\right]$, $B=\mathrm{const}$, admit holomorphically projective transformation (non-trivial for $B\ne 0$) [4,5].

Under the holomorphically projective mapping of $K_n\left[B\right]$ onto ${\overline{K}}_n\left[\overline{B}\right]$, the tensors ${\stackrel{*}{Z}}_{ijk}^h$ and ${\stackrel{*}{Z}}_{ij}$ are invariant with regard to this mapping, whereas

$${\stackrel{*}{Z}}_{ijk}^h\equiv R_{ijk}^h-B\left({\delta }_k^h{g}_{ij}-{\delta }_j^h{g}_{ik}-e{\delta }_{\overline{k}}^h{g}_{i\overline{j}}+e{\delta }_{\overline{j}}^h{g}_{i\overline{k}}-2e{\delta }_{\overline{i}}^h{g}_{j\overline{k}}\right);{\stackrel{*}{Z}}_{ij}\equiv {\stackrel{*}{Z}}_{ij\alpha }^{\alpha }.$$

We note that an analogue of Equation (5) is fulfilled for geodesic mappings between Einstein spaces, include spaces of constant curvature (see, e.g., [5,6,7,8,9,10,11,17]). Moreover, these equations characterize the $V\left(K\right)$ spaces by Solodovnikov [64] (for positive definite metrics) (see e.g. [9], p. 209) and $V_n\left(B\right)$ spaces by Mikeš [4,6,65] (for indefinite metrics).

5. Rigidity of Kähler Spaces’ Respective Holomorphically Projective Mappings “in Whole”

The investigation of the NHPM “in whole” of classical Kähler spaces (under the conditions of compactness or completness) were studied in [7,30,35,50,51,52,53,54].

5.1. Holomorphically Projective Mappings and Fundamental Functions Along Geodesics

Let us suppose that f is a holomorphically projective mapping of Kähler space $K_n$ onto Kähler space ${\overline{K}}_n$ and Equation (4) holds with ${\psi }_i={\partial }_i\mathrm{\Psi }$; B and $\overline{B}$ are constants. Let $\gamma \left(s\right)$ be a geodesic on $K_n$ and a corresponding analytically planar curve $\overline{\gamma }\left(\tau \left(s\right)\right)$ on ${\overline{K}}_n$ with natural parameter s and with canonical parameter $\tau$, respectively. Assume $\dot{\tau }=d\tau \left(s\right)/ds>0$ for the parameter transformation $\tau =\tau \left(s\right)$. The following holds:

$$g_{ij}{\dot{\gamma }}^i{\dot{\gamma }}^j=\epsilon =\pm 1,0\mathrm{and}{\overline{g}}_{ij}{\dot{\gamma }}^i{\dot{\gamma }}^j=cexp\left(4\mathrm{\Psi }\left(s\right)\right),c=\mathrm{const}.$$

(6)

The first equality is generally known, and the second follows from the contraction of (2) with ${\dot{\gamma }}^i{\dot{\gamma }}^j{\dot{\gamma }}^k$.

By differentiating $\gamma \left(s\right)=\overline{\gamma }\left(\tau \left(s\right)\right)$ with respect to parameter s, we obtain

$$\dot{\gamma }\left(s\right)=\stackrel{\circ }{\overline{\gamma }}\left(\tau \left(s\right)\right)·\dot{\tau }\left(s\right),\mathrm{where}\stackrel{\circ }{\overline{\gamma }}=d\overline{\gamma }\left(\tau \right)/d\tau .$$

(7)

Since $\tau$ is canonical of $\overline{\gamma }$, we can obtain $\overline{g}(\stackrel{\circ }{\overline{\gamma }},\stackrel{\circ }{\overline{\gamma }})=\overline{c}(=\mathrm{const}),$ and it follows that $\overline{g}(\dot{\gamma },\dot{\gamma })=\overline{c}·{\dot{\tau }}^2$.

Via a direct calculation that the geodesic $\gamma \left(s\right)$ with equation ${\nabla }_{\dot{\gamma }\left(s\right)}\dot{\gamma }\left(s\right)=0$ is mapped to an analytically planar curve $\overline{\gamma }\left(\tau \left(s\right)\right)$ with equation ${\overline{\nabla }}_{\stackrel{\circ }{\overline{\gamma }}\left(\tau \right)}\stackrel{\circ }{\overline{\gamma }}\left(\tau \right)={\varrho }_2\left(\tau \right)·\stackrel{\circ }{\overline{\gamma }}\left(\tau \right)$, we obtain the following formula:

$$\dot{\tau }\left(s\right)=\tilde{c}·e^{2\mathrm{\Psi }},\tilde{c}>0.$$

(8)

Along the geodesic $\gamma \left(s\right)$, we will set $\mathrm{\Psi }\left(s\right)=\mathrm{\Psi }\left(\gamma \right(s\left)\right)$ and from it, $\dot{\mathrm{\Psi }}\left(s\right)={\psi }_{\alpha }{\dot{\gamma }}^{\alpha }$. For the tensor ${\psi }_{ij}$ $(\equiv {\psi }_{i,j}-{\psi }_i{\psi }_j-e{\psi }_{\overline{i}}{\psi }_{\overline{j}})$, the equality ${\psi }_{ij}={\psi }_{ji}=-e{\psi }_{\overline{i}\overline{j}}$ holds. From this, it follows that tensor ${\psi }_{\overline{i}j}$ is skew symmetric, and ${\psi }_{\overline{\alpha }\beta }{\dot{\gamma }}^{\alpha }{\dot{\gamma }}^{\beta }=0$. From the definition of ${\psi }_{ij}$, it is evident that

$${\psi }_{\overline{i},j}={\psi }_{\overline{i}}{\psi }_j-{\psi }_i{\psi }_{\overline{j}}+{\psi }_{\overline{i}j}.$$

By differentiating the expression ${\psi }_{\overline{\alpha }}{\dot{\gamma }}^{\alpha }$ with respect to s, from the above, we make sure that

${\left({\psi }_{\overline{\alpha }}{\dot{\gamma }}^{\alpha }\right)}^{.}=2\dot{\mathrm{\Psi }}·\left({\psi }_{\overline{\alpha }}{\dot{\gamma }}^{\alpha }\right)$; moreover, after integrating it is obvious that

$${\psi }_{\overline{\alpha }}{\dot{\gamma }}^{\alpha }=\chi ·e^{2\mathrm{\Psi }},\mathrm{where}\chi \mathrm{is}\mathrm{constant}.$$

5.2. HP Mappings with ${\psi }_{ij}=\overline{B}{\overline{g}}_{ij}-B{g}_{ij}$ and Fundamental Functions Along Geodesics

Next, we will study the holomorphically projective mapping where Condition (4) is valid, i.e., ${\psi }_{ij}=\overline{B}{\overline{g}}_{ij}-B{g}_{ij}$, where B and $\overline{B}$ are constants. If this mapping is non-trivial, then the spaces $K_n$ and ${\overline{K}}_n$ will be $K_n\left[B\right]$ and ${\overline{K}}_n\left[\overline{B}\right]$, respectively.

We can write Condition (4) in an expanded form as follows:

$${\psi }_{i,j}={\psi }_i{\psi }_j+e{\psi }_{\overline{i}}{\psi }_{\overline{j}}+\overline{B}{\overline{g}}_{ij}-B{g}_{ij}.$$

(9)

We calculate $\ddot{\mathrm{\Psi }}\left(s\right)$ according to geodesics $\gamma \left(s\right)$: $\ddot{\mathrm{\Psi }}\left(s\right)={\left(\dot{\mathrm{\Psi }}\right)}^{.}={\left({\psi }_{\alpha }{\dot{\gamma }}^{\alpha }\right)}^{.}={\psi }_{\alpha ,\beta }{\dot{\gamma }}^{\alpha }{\dot{\gamma }}^{\beta }$. Moreover, after using (9), we obtain

$$\ddot{\mathrm{\Psi }}={\left(\dot{\mathrm{\Psi }}\right)}^2+b·e^{4\mathrm{\Psi }}-a,$$

(10)

where $a=\epsilon B$ and $b=c\overline{B}+e{\chi }^2$.

We substitute $q=e^{-2\mathrm{\Psi }\left(s\right)}$. Then, Equation (9) is equivalent to

$$2q\ddot{q}={\dot{q}}^2-4b+4a{q}^2.$$

(11)

The derivative of (11) gives the equation $\stackrel{⃛}{q}=4a\dot{q}$, which has a solution

$$\begin{array}{cc}\left(\mathrm{a}\right)q=c_0+c_{1}s+c_2{s}^2,\hfill & \mathrm{if}a=0,\hfill \\ \left(\mathrm{b}\right)q=c_0+c_{1}cosh\left(\alpha s\right)+c_{2}sinh\left(\alpha s\right),\hfill & \mathrm{if}a>0,\hfill \\ \left(\mathrm{c}\right)q=c_0+c_{1}cos\left(\alpha s\right)+c_{2}sin\left(\alpha s\right),\hfill & \mathrm{if}a<0,\hfill \end{array}$$

(12)

where $\alpha =2\sqrt{\left|a\right|}$, and $c_0,c_1,c_2$ are constants. Since the function q must satisfy Equation (11), $c_i$ coefficients are tied to each other.

We will analyze the obtained results in terms of compactness and completeness of the studied geodesics $\gamma$ and their image $\overline{\gamma }=f\left(\gamma \right)$:

Lemma 4.

The function $\mathrm{\Psi }\left(s\right)$ is constant for $a\equiv B·g(\dot{\gamma },\dot{\gamma })\ge 0$

(a)

If geodesic γ on $K_n$ is compact;

(b)

If geodesic γ on $K_n$ and its image $\overline{\gamma }$ on ${\overline{K}}_n$ are complete;

(c)

If geodesic γ on $K_n$ is complete, and $a\equiv B·g(\dot{\gamma },\dot{\gamma })=0$ and $b\equiv \overline{B}c+e{\chi }^2=0$.

Proof. 

Since $a\ge 0$ and $\mathrm{\Psi }\left(s\right)=-1/2ln\left(q\right(s\left)\right)$, it is enough to analyze Formulas (a) and (b) in (12).

(a) The proof of Case (a) is trivial because the non-constant function $\mathrm{\Psi }\left(s\right)$ is not bounded for $s\in \mathbb{R}$.

(b) The proof of the analogue of Case (b) has been completed by many authors and relies on ideas by Couty [51] in an investigation of projective transformations of Einstein manifolds, and by Shen [66] in an investigation of Finsler Einstein geodesically equivalent metrics.

Since $q=exp(-2\mathrm{\Psi }(s\left)\right)$, it follows from (8) that $\dot{\tau }\left(s\right)=\tilde{c}exp\left(2\mathrm{\Psi }\right)=\tilde{c}/q\left(s\right)>0$. Thus, $\tau \left(s\right)={\int }_{s_0}^s\tilde{c}/q\left(t\right)dt$.

For functions $q=c_0+c_{1}s+c_2{s}^2$ and $q=c_0+c_{1}cosh\left(\alpha s\right)+c_{2}sinh\left(\alpha s\right)$, $(c_1\ne 0$ or $c_2\ne 0)$, this integral diverges (goes to infinity in finite time s). Then, $c_1=c_2=0$ and $\tau =\mathrm{const}·s+s_0$, and it follows that $\dot{\tau }=\mathrm{const}$. Evidently, the function $\mathrm{\Psi }\left(s\right)$ is constant along geodesic $\gamma \left(s\right)$.

(c) The proof of Case (c) is trivial because for non-constant function $q\left(s\right)$, $s_0$ exists, for which $q\left(s_0\right)=0$. This is the contradiction with $q\left(s\right)>0$ for $s\in \mathbb{R}$.

5.3. Holomorphically Projective Mappings of $K_n$[0] with n Complete Geodesics

Holomorphically projective mapping $K_n$[0] onto ${\overline{K}}_n\left[\overline{B}\right]$ is characterized by Equation (2) and the following equation:

$${\psi }_{ij}\equiv {\psi }_{i,j}-{\psi }_i{\psi }_j+{\psi }_{\overline{i}}{\psi }_{\overline{j}}=\overline{B}{\overline{g}}_{ij},\overline{B}=\mathrm{const}.$$

(13)

We prove the following theorem:

Theorem 2.

Let $K_n$ admit holomorphically projective mapping onto ${\overline{K}}_n$ by Formula (13). Simultaneously, let us consider that through a given point and given linearly independent directions, n complete geodesics pass that satisfy Lemma 4. If one of the following conditions hold then this mapping is trivial (affine):

(a) $\overline{B}=0$;

(b) A geodesic γ has a non-isotropic image $f\left(\gamma \right)$;

(c) The above point is non-planar and around it $g\in C^3$.

Proof. 

Let the conditions of the theorem be satisfied. Then, according to the given geodesics, the function $\mathrm{\Psi }\left(s\right)$ is constant. Thus, at point $x_0$ in the direction of these geodesics, ${\partial }_{\alpha }\mathrm{\Psi }\left(x_0\right){\dot{\gamma }}^{\alpha }={\psi }_{\alpha }\left(x_0\right){\dot{\gamma }}^{\alpha }=0$ is vanishing in the tangent directions. Therefore, ${\psi }_i\left(x_0\right)=0$ must hold there.

(a) In Case (a), Equation (13) has the form ${\psi }_{i,j}={\psi }_i{\psi }_j-{\psi }_{\overline{i}}{\psi }_{\overline{j}}$. This equation for initial conditions $\psi \left(x_0\right)={\psi }_0$ and ${\psi }_i\left(x_0\right)=0$ only has a trivial solution $\psi \left(x\right)={\psi }_0$. Moreover, ${\psi }_i\left(x\right)=0$, and this mapping is trivial (affine).

(b) If $f\left(\gamma \right)$ is not isotropic, then after contracting (13) with ${\dot{\gamma }}^i{\dot{\gamma }}^j$, we obtain $\overline{B}·\overline{g}(\dot{\gamma },\dot{\gamma })=0$. From this, we obtain $\overline{B}=0$, and this case leads to Case (a).

(c) We note that Equation (13) is equivalent to the equation ${\lambda }_{i,j}=\mu g_{ij}$, where $\mu$ is constant. The integrability conditions of these equations have the form ${\lambda }_{\alpha }{R}_{ijk}^{\alpha }$=0. We covariantly differentiate them and use the following: $\mu R_{lijk}+{\lambda }_{\alpha }{R}_{ijk,l}^{\alpha }=0$. From ${\psi }_i\left(x_0\right)$, it follows that ${\lambda }_i\left(x_0\right)=0$. Moreover, because $R_{hijk}\left(x_0\right)\ne 0$, we obtain $\mu =0$.

Therefore, equation ${\lambda }_{i,j}=0$ for initial condition ${\lambda }_i\left(x_0\right)=0$ only has a trivial solution ${\lambda }_i\left(x\right)=0$. It follows that ${\psi }_i\left(x\right)$ is vanishing, and the holomorphically projective mapping is trivial (i.e., affine). □

Note that in the above assumption, $K_n$[0] and ${\overline{K}}_n$[$\overline{B}$] do not exist, as they are holomorphically projective-correspondent.

5.4. Holomorphically Projective Mappings of $K_n$[B] with Finite Complete Geodesics

Theorem 3.

Let $K_n$ admit holomorphically projective mapping onto ${\overline{K}}_n$ with Formula (5) with $B=$ const. $\ne 0$. Simultaneously, let us consider that through a given point $x_0$ (in which $P_{ijk}^h\left(x_0\right)\ne 0$ and around it $\overline{g}\in C^3$), and given linearly independent directions, there pass n complete geodesics that satisfy Lemma 4. If one of the following conditions holds, then this mapping is trivial (affine):

(a) $\overline{B}=0$;

(b) A geodesic γ is isotropic, and it has a non-isotropic image $f\left(\gamma \right)$.

Proof. 

Let the conditions of the theorem be satisfied. Then, according to the given geodesics, the function $\mathrm{\Psi }\left(s\right)$ is constant. Thus, at point $x_0$ (in which $P_{ijk}^h\ne 0$) in the direction of these geodesics, ${\partial }_{\alpha }\mathrm{\Psi }\left(x_0\right){\dot{\gamma }}^{\alpha }={\psi }_{\alpha }\left(x_0\right){\dot{\gamma }}^{\alpha }=0$ is vanishing in the tangent directions. Therefore, ${\psi }_i\left(x_0\right)=0$ must hold there.

(a) This case is analogous to Case (c) in Theorem 2. The point $x_0$ is non-planar, but in a ${\overline{K}}_n$ space. Here, i.e., in the space ${\overline{K}}_n$, the equations of convergent vector fields ${\overline{\lambda }}_i$ also hold, ${\overline{\lambda }}_{i|j}=\mu {\overline{g}}_{ij}$, where $\mu$ is constant, and “ | ” is a covariant derivative on ${\overline{K}}_n$. Since there are ${\overline{\lambda }}_i\left(x_0\right)=0$ in point $x_0$, ${\overline{\lambda }}_i\left(x\right)=0$ for any point.

(b) If $\gamma$ is isotropic and its image $f\left(\gamma \right)$ is non-isotropic, then after contracting (13) with ${\dot{\gamma }}^i{\dot{\gamma }}^j$, we obtain $\overline{B}·\overline{g}(\dot{\gamma },\dot{\gamma })=0$. From this, we obtain $\overline{B}=0$, and this case leads to Case (a). □

6. Summary

The main results of our study are Theorems 2 and 3. These results show that for a mapping to be rigid, the space does not necessarily need to be complete. It is sufficient for a finite number of geodesics and their images to be complete. In other words, the space is uniquely determined by a specific set of geodesics, which constitute the supporting structure (skeleton) of the space.

In practical terms, the space is uniquely determined by the given geodesics, which serve as the structural skeleton (reinforcement) of the surface.