Geometric Inequalities for a Submanifold Equipped with Distributions

Abstract

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.

1. Introduction

The main message of the theory of embedding surfaces in three-dimensional Euclidean space is as follows. Surfaces with positive curvature are easily embedded in 3D space (A.D. Aleksandrov and A.V. Pogorelov), whereas surfaces with negative curvature usually do not allow for such embedding (D. Gilbert and N.V. Efimov). The famous embedding theorem of J.F. Nash [1] gave hope for the use of extrinsic geometry when a Riemannian n-dimensional manifold is considered as smoothly isometrically embedded in a Euclidean m-space, but it does not indicate the relationship between m and n in light of curvature invariants. The difficulty was in understanding smooth submanifolds (for $C^1$-immersions, the problem looks different, [2]) of a very large codimension using only a few known relationships (fundamental Gauss–Codazzi–Ricci equations) between intrinsic and extrinsic geometry.

A further development of the extrinsic geometry of submanifolds (and, more recently, of foliations; see [3]) led to the following problem (see [4] (Problem 2)): find a simple optimal connection between the intrinsic and extrinsic invariants of a submanifold in a Riemannian manifold; in particular, in a real space form. The imposition of the minimum condition for a submanifold in Euclidean space yields, e.g., the negativity of its Ricci curvature. This led to a question posed by S.S. Chern in 1968 on other obstacles for a Riemannian manifold to admit an isometric minimal immersion in a Euclidean space. To study these questions, it is necessary to introduce new types of Riemannian invariants and to find optimal relations between them and extrinsic invariants of submanifolds.

B.Y. Chen introduced the concept of $\delta$-curvature invariants for Riemannian manifolds in the 1990s and proved an optimal inequality for a submanifold, including $\delta$-curvature invariants and the square of mean curvature, e.g., [5], the case of equality led to the notion of “ideal immersions” in Euclidean space (isometric immersions with the smallest possible tension). The Chen invariants are obtained from the scalar curvature (which is the “sum” of sectional curvatures) by discarding some of the sectional curvatures. Some similar invariants are known for Kähler, para-Kähler, contact and affine manifolds, warped products and submersions; see surveys [6,7].

Distributions on a manifold, i.e., sub-bundles of the tangent bundle, arise in such topics as foliations (i.e., integrable distributions), submersions, Lie groups actions, almost product manifolds, contact manifolds and in physics, e.g., [3,8]. In [9], we introduced curvature invariants (different from Chen invariants) of a Riemannian manifold equipped with complementary orthogonal distributions, and proved the geometric inequality for submanifolds that includes our curvature invariants and the square of mean curvature. These curvature invariants are related to mixed scalar curvature: a well-known curvature invariant for foliations and Riemannian almost k-product manifolds, and, in particular, multiply twisted or warped products; see, e.g., [3].

In this article, we introduce more general invariants of a Riemannian manifold than in [9], which are related to the mutual curvature of noncomplementary pairwise orthogonal subspaces of the tangent bundle. Using invariants based on mutual curvatures, we prove geometrical inequalities for Riemannian submanifolds and give applications for sub-Riemannian submanifolds. Namely, we consider a smooth manifold equipped with a distribution $\mathcal{D}$. If $\mathcal{D}$ is non-integrable, we obtain a non-holonomic manifold, and if $\mathcal{D}$ is integrable (involutive), we obtain a foliated manifold, e.g., [3]. Recall that a non-holonomic manifold endowed with a Riemannian metric on $\mathcal{D}$ (which may be the restriction of a metric given on a tangent bundle) is the central object of sub-Riemannian geometry, e.g., [8]. We supplement the above problem with the following: find a simple optimal connection between the intrinsic and extrinsic invariants of a sub-Riemannian manifold isometrically immersed in an adapted way into another sub-Riemannian manifold; for example, ${\mathbb{R}}^m={\mathbb{R}}^{m_1}\times {\mathbb{R}}^{m_2}$ with a Euclidean metric (i.e., the image of $\mathcal{D}$ in ${\mathbb{R}}^m$ is parallel to ${\mathbb{R}}^{m_1}\times \left\{0\right\}$). The imposition of the $\mathcal{D}$-minimum condition leads to the following question: Given a sub-Riemannian manifold, what are the necessary conditions for M to admit a $\mathcal{D}$-minimal adapted isometric immersion in ${\mathbb{R}}^m={\mathbb{R}}^{m_1}\times {\mathbb{R}}^{m_2}$?

The article is organized as follows.

In Section 2, following the introductory Section 1, we define the mutual curvature of $k\ge 2$ pairwise orthogonal subspaces $V_1,\dots ,V_k$ of $T_{x}M$ at a point $x\in M$ as the sum of sectional curvatures of the planes nontrivially intersecting any two subspaces. In the case of one-dimensional subspaces, the mutual curvature is equal to half the scalar curvature of the subspace ${⨁}_i{V}_i$, and in the case of ${⨁}_i{V}_i=T_{x}M$, this is the mixed scalar curvature. Then, we introduce mutual curvature invariants of a Riemannian manifold and compare them with the classical Chen’s invariants.

In Section 3, with the main results, we prove Theorems 1 and 2 with geometric inequalities involving invariants based on mutual curvature and the square of mean curvature for a Riemannian submanifold. We also present corollaries on the absence of minimal isometric immersions and a proposition in the form of Chen’s maximum principle.

Section 4 contains three kinds of applications of the main results. First, we introduce invariants based on the mutual curvature of a sub-Riemannian manifold and compare them with the corresponding Chen-type invariants, whose theory is similar to the theory of Chen invariants. Then, geometric inequalities are proved for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions. This gives us sufficient conditions for the absence of $\mathcal{D}$-minimal immersions of sub-Riemannian manifolds in Euclidean space.

2. Invariants of a Riemannian Manifold Based on Mutual Curvature

Let $(M,g)$ be an n-dimensional Riemannian manifold with the Levi-Civita connection ∇ and the curvature tensor $R_{X,Y}={\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$. The Ricci tensor ${Ric}_{X,Y}=trace(Z\mapsto R_{Z,X}Y)$ is a contraction of R. The scalar curvature $\tau ={trace}_{g}Ric$ of M is the trace of the Ricci tensor, e.g., [10]. Some authors, e.g., [5,6,7], define the scalar curvature as half of “trace Ricci”.

Let $V_1,\dots ,V_k(k\ge 2)$ be mutually orthogonal subspaces of $T_{x}M$ at a point $x\in M$ with $dim{V}_i=n_i\ge 1$. Let $\left\{e_i\right\}$ be an orthonormal frame of a subspace $V={⨁}_{i=1}^k{V}_i$ such that $\{e_1,\dots ,e_{n_1}\}\subset V_1$, …, $\{e_{n_{k-1}+1},\dots ,e_{n_k}\}\subset V_k$. The mutual curvature of a set $\{V_1,\dots ,V_k\}$ is defined by

$${\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)={\sum }_{i<j}{\sum }_{n_{i-1}<a\le n_i,n_{j-1}<b\le n_j}K(e_a\wedge e_b),$$

(1)

where $K(e_a\wedge e_b)=g(R_{e_a,e_b}{e}_b,e_a)$ is the sectional curvature of the plane $e_a\wedge e_b$.

Note that ${\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)$ does not depend on the choice of frames. We immediately have

$${\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)={\sum }_{i<j}{\mathrm{S}}_{\mathrm{m}}(V_i,V_j),$$

where the mutual curvature of the pair $(V_i,V_j)$ is given by

$${\mathrm{S}}_{\mathrm{m}}(V_i,V_j)={\sum }_{n_{i-1}<a\le n_i,n_{j-1}<b\le n_j}K(e_a\wedge e_b).$$

For the scalar curvature $\tau \left(V\right)={trace}_g{Ric|}_V$ (the trace of the Ricci tensor on a subspace $V={⨁}_{i=1}^k{V}_i$), we obtain

$$\tau \left(V\right)=2{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)+{\sum }_{i=1}^k\tau \left(V_i\right),$$

(2)

where $\tau \left(V_i\right)={trace}_g{Ric|}_{V_i}$ are scalar curvatures of subspaces $V_i$. Thus, if all subspaces $V_i$ are one-dimensional, then $2{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)=\tau \left(V\right)$.

For an integer $k\ge 2$, denote by $S(n,k)$ the set of unordered k-tuples $(n_1,\dots ,n_k)$ of natural numbers satisfying $n_1+\dots +n_k\le n$. Denote by $S\left(n\right)$ the set of all unordered k-tuples with $k\ge 2$.

Definition 1.

For $(n_1,\dots ,n_k)\in S\left(n\right)$, the functions ${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)$ and ${\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)$ are defined by

$${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\left(x\right)=max{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k),{\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)\left(x\right)=min{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k),$$

(3)

where $V_1,\dots ,V_k$ run over all k mutually orthogonal subspaces of $T_{x}M$ with $dim{V}_i=n_i(i=1,\dots ,k)$.

If the sectional curvature K of $(M,g)$ satisfies $c\le K\le C$ and ${\sum }_{i=1}^k{n}_i=s\le n$, then

$$\begin{array}{c}\hfill \frac{c}{2}(s^2-{\sum }_i{n}_i^2)=c{\sum }_{i<j}{n}_i{n}_j\le {\delta }_{\mathrm{m}}^{-}(n_1,\dots n_k)\le \\ \hfill \le {\delta }_{\mathrm{m}}^{+}(n_1,\dots n_k)\le C{\sum }_{i<j}{n}_i{n}_j=\frac{C}{2}(s^2-{\sum }_i{n}_i^2).\end{array}$$

(4)

Example 1.

For a subspace V spanned by $q+1$ orthonormal vectors $\{e_0,e_1,\dots ,e_q\}$ of $(M,g)$, the q-th Ricci curvature is ${Ric}_q\left(V\right)={\sum }_{i=1}^{q}K(e_0,e_i)$; see [11]. For $k=2$, using the intermediate Ricci curvature, we obtain ${\delta }_{\mathrm{m}}^{+}(1,n_2)\left(x\right)=max{Ric}_{n_2}\left(V\right)$ and ${\delta }_{\mathrm{m}}^{-}(1,n_2)\left(x\right)=min{Ric}_{n_2}\left(V\right)$, where $V=span(V_1,V_2)$ and $V_1,V_2$ run over all mutually orthogonal subspaces of $T_{x}M$ such that $dim{V}_1=1$ and $dim{V}_2=n_2$.

For a k-tuple $(k\ge 0)$ and $x\in M$, B.-Y Chen defined the following invariants, e.g., [5] (Section 13.2):

$$\begin{array}{c}\hfill 2\delta (n_1,\dots ,n_k)\left(x\right)=\tau \left(x\right)-min\{\tau \left(V_1\right)+\dots +\tau \left(V_k\right)\},\\ \hfill 2\widehat{\delta }(n_1,\dots ,n_k)\left(x\right)=\tau \left(x\right)-max\{\tau \left(V_1\right)+\dots +\tau \left(V_k\right)\},\end{array}$$

(5)

where $V_1,\dots ,V_k$ run over all k mutually orthogonal subspaces of $T_{x}M$ with $dim{V}_i=n_i(i=1,\dots ,k)$, and the coefficient 2 is due to the definition of the scalar curvature in [5] as half of the “trace Ricci". Our ${\delta }_{\mathrm{m}}$-invariants are related to the Chen invariants (5) by the following inequalities.

Proposition 1.

Let $k\ge 2$. Then, for $n_1+\dots +n_k<n$, the following inequalities are valid:

$$\begin{array}{c}\hfill {\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\ge \delta (n_1,\dots ,n_k)-\delta (n_1+\dots +n_k),\\ \hfill {\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)\le \widehat{\delta }(n_1,\dots ,n_k)-\widehat{\delta }(n_1+\dots +n_k).\end{array}$$

(6)

If $n_1+\dots +n_k=n$, then $\widehat{\delta }(n_1,\dots ,n_k)={\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)\le {\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)=\delta (n_1,\dots ,n_k)$, and if $n_1+\dots +n_k=n-1$, then

$$\widehat{\delta }(n_1,\dots ,n_k)-minRic\ge {\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)\ge {\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\ge \delta (n_1,\dots ,n_k)-maxRic.$$

Proof. 

Using (2) and the equality $mina=-max(-a)$, we obtain

$$\begin{array}{ccc}& & 2\delta (n_1,\dots ,n_k)\left(x\right)=\tau \left(V\right)-min\{\tau \left(V_1\right)+\dots +\tau \left(V_k\right)\}\hfill \\ & & =\tau \left(x\right)+max({\tau }_k\left(x\right)-(\tau \left(V_1\right)+\dots +\tau \left(V_k\right))-{\tau }_k\left(V\right))\hfill \\ & & \le \tau \left(x\right)-min{\tau }_k\left(x\right)+2max{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)\hfill \\ & & =2\delta (n_1+\dots +n_k)\left(x\right)+2{\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\left(x\right),\hfill \end{array}$$

Hence, (6)${}_1$ is valid. The proof of (6)${}_2$ is similar. The case of $n_1+\dots +n_k=n$ follows from (6). The case of $n_1+\dots +n_k=n-1$ follows from $\delta (n-1)\left(x\right)=maxRic\left(x\right)$ and $\widehat{\delta }(n-1)\left(x\right)=minRic\left(x\right)$. □

Corollary 1.

If $(M,g)$ has a non-negative sectional curvature and $k\ge 2$, then

$$\widehat{\delta }(n_1,\dots ,n_k)\le {\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)\le {\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\le \delta (n_1,\dots ,n_k),$$

and if $(M,g)$ has a nonpositive sectional curvature, then the inequalities are opposite.

3. Main Results

Let $f:(M,g)\to (\overline{M},\overline{g})$ be an isometric immersion of an n-dimensional Riemannian manifold into another Riemannian manifold of dimension m. We will identify M with its image $f\left(M\right)$ (since the induced metric on $f\left(M\right)$ is equal to g) and put a top “bar” for objects related to $\overline{M}$. Let $h:TM\times TM\to T{M}^{\perp }$ be the second fundamental form of the immersion f, where $T{M}^{\perp }$ is the normal bundle of the submanifold $f\left(M\right)$. Recall the Gauss equation for an isometric immersion f, e.g., [5]:

$$\overline{g}({\overline{R}}_{Y,Z}U,X)=g(R_{Y,Z}U,X)+\overline{g}(h(Y,U),h(Z,X))-\overline{g}(h(Z,U),h(Y,X)),$$

(7)

where $U,X,Y,Z\in TM$ and $\overline{R}$ and R are the curvature tensors of $(\overline{M},\overline{g})$ and $(M,g)$, respectively. The mean curvature vector of a subspace $V\subset T_{x}M$ is given by $H_V={\sum }_{i}h(e_i,e_i)$, where $e_i$ is an orthonormal basis of V. Set

$${\mathcal{H}}_x\left(s\right)=max\{\parallel H_V\parallel :V\subset T_{x}M,dimV=s>0\}.$$

(8)

If $s=n$, then obviously $H_V=H\left(x\right)$. Note that, for $s<n$, the condition $\mathcal{H}\left(s\right)=0$ implies that the submanifold is totally geodesic ($h=0$). An isometric immersion f is called mixed totally geodesic on $V={⨁}_{i=1}^k{V}_i$ if $h(X,Y)=0$ for all $X\in V_i,Y\in V_j$ and $i\ne j$.

We obtain the following geometric inequality.

Theorem 1.

For any natural numbers $n_1,\dots ,n_k$ such that ${\sum }_i{n}_i=s\le n$, we obtain

$${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\le {\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)+\frac{k-1}{2k}\left\{\begin{array}{cc}\mathcal{H}{\left(s\right)}^2,& \mathrm{if}s<n,\\ {\parallel H\parallel }^2,& \mathrm{if}s=n,\end{array}\right.$$

(9)

where ${\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)$ are defined for $(\overline{M},\overline{g})$ similarly to ${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)$ for $(M,g)$. The equality in (9) holds at a point $x\in M$ if and only if there exist mutually orthogonal subspaces $V_1,\dots ,V_k$ of $T_{x}M$ with ${\sum }_i{n}_i=s\le n$ such that f is mixed totally geodesic on $V={⨁}_{i=1}^k{V}_i$, $H_1=\dots =H_k$, $\parallel H_V\parallel ={\mathcal{H}}_x\left(s\right)$ and ${\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_k)={\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\left(x\right)$.

Proof. 

Taking a trace of the Gauss equation (7) for the immersion f along V and $V_i$ yields the equalities

$$\begin{array}{ccc}\hfill \overline{\tau }\left(V\right)-\tau \left(V\right)& =& \parallel h_V{\parallel }^2-{\parallel H_V\parallel }^2,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \overline{\tau }\left(V_i\right)-\tau \left(V_i\right)& =& \parallel h_i{\parallel }^2-{\parallel H_i\parallel }^2,\hfill \end{array}$$

(11)

where $\overline{\tau }\left(V\right)$, $\overline{\tau }\left(V_i\right)$ and $\tau \left(V\right)$, $\tau \left(V_i\right)$ are the scalar curvatures of subspaces $V={⨁}_{i=1}^k{V}_i$ and $V_i$ for the curvature tensors $\overline{R}$ and R, respectively, and $H_i$ is the mean curvature vector of $V_i$ at the point $x\in M$.

Assume that $H_V\ne 0$ is satisfied on an open set $U\subset M$. We complement over U an adapted local orthonormal frame $\{e_1,\dots ,e_n\}$ of $(M,g)$ with vector $e_{n+1}$ parallel to $H_V$. Using $H_V={\sum }_{i=1}^k{H}_i$ and the algebraic inequality $a_1^2+\dots +a_k^2\ge \frac{1}{k}{(a_1+\dots +a_k)}^2$ for real $a_i=\overline{g}(H_i,e_{n+1})$, we find

$${\sum }_i\parallel H_i{\parallel }^2\ge {\sum }_i\overline{g}{(H_i,e_{n+1})}^2\ge \frac{1}{k}{\parallel H_V\parallel }^2,$$

(12)

and the equality holds if and only if $H_1=\dots =H_k$. The above inequality is trivially satisfied for $H_V=0$; hence, it is valid on M. Set $\parallel h_{ij}^{\mathrm{mix}}{\parallel }^2={\sum }_{e_a\in V_i,e_b\in V_j}{\parallel h(e_a,e_b)\parallel }^2$ for $i\ne j$.

Note that

$$\parallel h_V{\parallel }^2={\sum }_i\parallel h_i{\parallel }^2+{\sum }_{i<j}\parallel h_{ij}^{\mathrm{mix}}{\parallel }^2\ge {\sum }_i{\parallel h_i\parallel }^2,$$

(13)

and the equality holds if and only if $\parallel h_{ij}^{\mathrm{mix}}{\parallel }^2=0(\forall i<j)$, i.e., f is mixed totally geodesic along V.

By (10)–(13) and the following equalities, see (2):

$$\overline{\tau }\left(V\right)=2{\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_k)+{\sum }_i\overline{\tau }\left(V_i\right),\tau \left(V\right)=2{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)+{\sum }_i\tau \left(V_i\right),$$

we obtain

$$\begin{array}{ccc}\hfill & & 2{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_k)=2{\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_k)+{\sum }_i(\overline{\tau }\left(V_i\right)-\tau \left(V_i\right))+\parallel H_V{\parallel }^2-{\parallel h_V\parallel }^2\hfill \\ \hfill & & \le 2{\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)-(\parallel h_V{\parallel }^2-{\sum }_i\parallel h_i{\parallel }^2)+(\parallel H_V{\parallel }^2-{\sum }_i\parallel H_i{\parallel }^2)\hfill \\ & & \le 2{\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)+\frac{k-1}{k}\mathcal{H}{\left(s\right)}^2,\hfill \end{array}$$

and the equality holds in the second line if and only if ${\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_k)={\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)$ and $\parallel H_V\parallel ={\mathcal{H}}_x\left(s\right)$ at each point $x\in M$. This proves (9) for $s<n$. The case ${\sum }_i{n}_i=n$ of (9) was proved in [9]. □

Remark 1.

For isometric immersions of $(M,g)$ into $(\overline{M},\overline{g})$ with sectional curvature bounded above by real c, e.g., when $(\overline{M},\overline{g})$ is the real space form $\overline{M}\left(c\right)$, for ${\sum }_i{n}_i=s<n$ from (4) and (9), we obtain

$${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\le \frac{k-1}{2k}\mathcal{H}{\left(s\right)}^2+\frac{c}{2}(s^2-{\sum }_i{n}_i^2),$$

and for ${\sum }_i{n}_i=n$, from (9), we obtain the inequality

$${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\le \frac{k-1}{2k}{\parallel H\parallel }^2+\frac{c}{2}(n^2-{\sum }_i{n}_i^2),$$

(14)

whose right hand side coincides with the right hand side of [5] (Equation (13.43)) for ${\sum }_i{n}_i=n$:

$$\delta (n_1,\dots ,n_k)\le \frac{n^2(n+k-1-{\sum }_i{n}_i)}{2(n+k-{\sum }_i{n}_i)}{\parallel H\parallel }^2+\frac{c}{2}[n(n-1)-{\sum }_i{n}_i(n_i-1)].$$

(15)

Set ${\delta }_{\mathrm{m}}^{+}\left(k\right)=max{\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)$ and ${\delta }_{\mathrm{m}}^{-}\left(k\right)=min{\delta }_{\mathrm{m}}^{-}(n_1,\dots ,n_k)$, where $(n_1,\dots ,n_k)$ run over $S(n,k)$. The ${\overline{\delta }}_{\mathrm{m}}^{+}(k+1)$ are defined for $(\overline{M},\overline{g})$ similarly to ${\delta }_{\mathrm{m}}^{+}(k+1)$ for $(M,g)$.

The following geometric inequality involves the square of mean curvature and supplements (9).

Theorem 2.

For any $k\ge 2$, we obtain

$$\begin{array}{ccc}\hfill & & {\delta }_{\mathrm{m}}^{-}\left(k\right)\le \frac{k-1}{2k(k+1)}{\parallel H\parallel }^2+{\overline{\delta }}_{\mathrm{m}}^{+}(k+1).\hfill \end{array}$$

(16)

The equality in (16) holds at a point $x\in M$ if and only if there exist mutually orthogonal subspaces $V_1,\dots ,V_{k+1}$ of $T_{x}M$ with ${\sum }_{i=1}^{k+1}{n}_i=n$ such that f is mixed totally geodesic, $H_1=\dots =H_{k+1}$, ${\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_{k+1})={\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_{k+1})$ and ${\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,{\widehat{V}}_i,\dots ,V_{k+1})={\delta }_{\mathrm{m}}^{-}\left(k\right)$ for any $i=1,\dots ,k+1$, where ${\widehat{V}}_i$ means removing the space $V_i$ from the set $V_1,\dots ,V_{k+1}$.

Proof. 

Let $V_{k+1}$ be the orthogonal complement to $V={⨁}_{i=1}^k{V}_i$, i.e., ${⨁}_{i=1}^{k+1}{V}_i=T_{x}M$. Note that

$${\sum }_i{\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,{\widehat{V}}_i,\dots ,V_{k+1})=(k+1){\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_{k+1}).$$

We also obtain ${\delta }_{\mathrm{m}}^{-}\left(k\right)\le {\delta }_{\mathrm{m}}^{-}(n_1,\dots ,{\widehat{n}}_i,\dots ,n_{k+1})\le {\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,{\widehat{V}}_i,\dots ,V_{k+1})$ for any $i=1,\dots ,k+1$. Thus, ${\delta }_{\mathrm{m}}^{-}\left(k\right)\le {\mathrm{S}}_{\mathrm{m}}(V_1,\dots ,V_{k+1})$, and using (9) for ${\sum }_i{n}_i=n$ gives (16). □

From Theorems 1 and 2, we obtain the assertions on the absence of isometric immersions.

Corollary 2.

There are no minimal isometric immersions of a Riemannian manifold $(M^n,g)$ into Euclidean space ${\mathbb{R}}^m$ with any of the following properties:

(a) ${\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)>0$ for some $(n_1,\dots ,n_k)$ with ${\sum }_i{n}_i=n$, (b) ${\delta }_{\mathrm{m}}^{-}\left(k\right)>0$ for some $k\ge 2$.

For each k-tuple $(n_1,\dots ,n_k)$ with ${\sum }_i{n}_i=s\le n$, we define the normalized ${\delta }_{\mathrm{m}}$-curvature by

$${\Delta }_{\mathrm{m}}(n_1,\dots ,n_k)=\frac{2k}{k-1}{\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k),$$

and put ${\overline{\Delta }}_{\mathrm{m}}\left(s\right):=max{\Delta }_{\mathrm{m}}(n_1,\dots ,n_k)$, where $(n_1,\dots ,n_k)$ run over $S\left(n\right)$ and ${\sum }_i{n}_i=s$.

Corollary 3.

For every isometric immersion of a Riemannian n-manifold into a Euclidean space with arbitrary codimension, we have $\mathcal{H}{\left(s\right)}^2\ge {\overline{\Delta }}_{\mathrm{m}}\left(s\right)$ for any $s<n$, and ${\parallel H\parallel }^2\ge {\overline{\Delta }}_{\mathrm{m}}\left(n\right)$.

The following assertion (compare with the maximum principle in [5] (p. 268)) follows from Theorem 1.

Proposition 2.

If a submanifold $M^n$ of a Euclidean space satisfies $\mathcal{H}{\left(s\right)}^2={\Delta }_{\mathrm{m}}(n_1,\dots ,n_k)$ for some k-tuple $(n_1,\dots ,n_k)$ with ${\sum }_i{n}_i=s\le n$, then for every $(m_1,\dots ,m_k)$ with ${\sum }_i{m}_i=s$, we have ${\Delta }_{\mathrm{m}}(n_1,\dots ,n_k)\ge {\Delta }_{\mathrm{m}}(m_1,\dots ,m_k)$.

Proof. 

By conditions, ${\Delta }_{\mathrm{m}}(n_1,\dots ,n_k)={\overline{\Delta }}_{\mathrm{m}}\left(s\right)$. Since ${\Delta }_{\mathrm{m}}(m_1,\dots ,m_k)\le \mathcal{H}{\left(s\right)}^2$, we obtain the required inequality ${\Delta }_{\mathrm{m}}(m_1,\dots ,m_k)\le {\Delta }_{\mathrm{m}}(n_1,\dots ,n_k)$, which completes the proof. □

Remark 2.

The case of equality in Corollary 3 is of special interest. Such extremal immersions in Euclidean space can be compared to “ideal immersions" introduced by Chen’s in terms of δ-invariants, e.g., [5] (Definition 13.3).

4. Applications

Based on the questions posed in the Introduction, here, we will discuss applications of the method presented in the previous sections: those related to isometric immersions of a sub-Riemannian manifold $(M,g;\mathcal{D})$ and a Riemannian manifold endowed with mutually orthogonal distributions.

1. Let us define the mutual curvature invariants ${\delta }_{\mathrm{m},\mathcal{D}}^{+}$ of a sub-Riemannian manifold similarly as invariants ${\delta }_{\mathrm{m}}^{+}$ in (3) using subspaces $V_i$ tangent to $\mathcal{D}$. The function ${\mathcal{H}}_{\mathcal{D}}\left(s\right)$ is similar to $\mathcal{H}\left(s\right)$ in (8). It is defined using subspaces V from $\mathcal{D}$. Note that ${\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)\le {\delta }_{\mathrm{m}}^{+}(n_1,\dots ,n_k)$, and, for $s<d$, it follows from the condition ${\mathcal{H}}_{\mathcal{D}}\left(s\right)=0$ that the immersion f is totally geodesic along $\mathcal{D}$, i.e., $h_{|{\mathcal{D}}_x}=0$.

Theorem 3.

Let $f:(M,g;\mathcal{D})\to (\overline{M},\overline{g})$ be an isometric immersion, and ${\sum }_i{n}_i=s\le d$. Then,

$$\begin{array}{c}\hfill {\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)\le {\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)+\frac{k-1}{2k}\left\{\begin{array}{cc}{\mathcal{H}}_{\mathcal{D}}{\left(s\right)}^2,& \mathrm{if}s<d,\\ \parallel H_{\mathcal{D}}{\parallel }^2,& \mathrm{if}s=d.\end{array}\right.\end{array}$$

(17)

The equality in (17) holds at a point $x\in M$ if and only if there exist mutually orthogonal subspaces $V_1,\dots ,V_k$ of ${\mathcal{D}}_x$ with ${\sum }_i{n}_i=s$ such that f is mixed totally geodesic on $V={⨁}_{i=1}^k{V}_i$, $H_1=\dots =H_k$, $\parallel H_V\parallel ={\mathcal{H}}_{{\mathcal{D}}_x}\left(s\right)$ and ${\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_k)={\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\left(x\right)$.

Proof. 

This repeats the proof of Theorem 1 using $V_i\subset \mathcal{D}$. □

Remark 3.

For isometric immersions of $(M,g;\mathcal{D})$ in $(\overline{M},\overline{g})$ with sectional curvature bounded above by c— for example, $(\overline{M},\overline{g})$ is the real space form $\overline{M}\left(c\right)$, and ${\sum }_i{n}_i=s\le d$—from (17), we obtain the following generalization of (14):

$${\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)\le \frac{c}{2}(d^2-{\sum }_i{n}_i^2)+\frac{k-1}{2k}\left\{\begin{array}{cc}{\mathcal{H}}_{\mathcal{D}}{\left(s\right)}^2,& \mathrm{if}s<d,\\ \parallel H_{\mathcal{D}}{\parallel }^2,& \mathrm{if}s=d.\end{array}\right.$$

An isometric immersion $f:(M,g;\mathcal{D})\to (\overline{M},\overline{g})$ is called $\mathcal{D}$-minimal if $H_{\mathcal{D}}\equiv 0$.

From Theorem 3, we obtain the assertion about the absence of isometric immersions.

Corollary 4.

There are no $\mathcal{D}$-minimal isometric immersions of a sub-Riemannian manifold $(M,g;\mathcal{D})$ into Euclidean space ${\mathbb{R}}^m$ with ${\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)>0$ for some $(n_1,\dots ,n_k)$ with ${\sum }_i{n}_i=d$.

For each k-tuple $(n_1,\dots ,n_k)$ with ${\sum }_i{n}_i=d$, we define the normalized ${\delta }_{\mathrm{m},\mathcal{D}}$-curvature by

$${\Delta }_{\mathrm{m},\mathcal{D}}(n_1,\dots ,n_k)=\frac{2k}{k-1}{\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k),$$

and put ${\overline{\Delta }}_{\mathrm{m},\mathcal{D}}:=max{\Delta }_{\mathrm{m},\mathcal{D}}(n_1,\dots ,n_k)$, where $(n_1,\dots ,n_k)$ run over $S\left(d\right)$ and ${\sum }_i{n}_i=d$.

Corollary 5.

For every isometric immersion of a sub-Riemannian manifold into a Euclidean space with arbitrary codimension, we have $\parallel H_{\mathcal{D}}{\parallel }^2\ge {\overline{\Delta }}_{\mathrm{m},\mathcal{D}}$.

The following assertion (compared with Proposition 2) follows immediately from Theorem 3.

Proposition 3.

If a non-holonomic submanifold $(M,\mathcal{D})$ of a Euclidean space satisfies ${\mathcal{H}}_{\mathcal{D}}{\left(s\right)}^2={\Delta }_{\mathrm{m},\mathcal{D}}(n_1,\dots ,n_k)$ for some k-tuple $(n_1,\dots ,n_k)$ with ${\sum }_i{n}_i=s\le d$, then for every $(m_1,\dots ,m_k)$ with ${\sum }_i{m}_i=s$, we have

$${\Delta }_{\mathrm{m},\mathcal{D}}(n_1,\dots ,n_k)\ge {\Delta }_{\mathrm{m},\mathcal{D}}(m_1,\dots ,m_k).$$

Proof. 

This is similar to the proof of Proposition 2. □

Definition 2.

We define the ${\delta }_{\mathcal{D}}$-invariants of a sub-Riemannian manifold similarly to (5) as:

$$\begin{array}{c}\hfill 2{\delta }_{\mathcal{D}}(n_1,\dots ,n_k)\left(x\right)={\tau }_{|\mathcal{D}}\left(x\right)-min\{\tau \left(V_1\right)+\dots +\tau \left(V_k\right)\},\\ \hfill 2{\widehat{\delta }}_{\mathcal{D}}(n_1,\dots ,n_k)\left(x\right)={\tau }_{|\mathcal{D}}\left(x\right)-max\{\tau \left(V_1\right)+\dots +\tau \left(V_k\right)\},\end{array}$$

(18)

where $V_1,\dots ,V_k$ run over all k mutually orthogonal subspaces of ${\mathcal{D}}_x$ with $dim{V}_i=n_i(i=1,\dots ,k)$.

The ${\delta }_{\mathrm{m},\mathcal{D}}^{\pm }$-invariants are related to the curvature invariants in (18) by the following inequalities.

Proposition 4.

Let $k\ge 2$. Then, for $n_1+\dots +n_k<d$, the following inequalities are valid:

$$\begin{array}{c}\hfill {\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)\ge {\delta }_{\mathcal{D}}(n_1,\dots ,n_k)-{\delta }_{\mathcal{D}}(n_1+\dots +n_k),\\ \hfill {\delta }_{\mathrm{m},\mathcal{D}}^{-}(n_1,\dots ,n_k)\le {\widehat{\delta }}_{\mathcal{D}}(n_1,\dots ,n_k)-{\widehat{\delta }}_{\mathcal{D}}(n_1+\dots +n_k),\end{array}$$

and, if $n_1+\dots +n_k=d$, then ${\widehat{\delta }}_{\mathcal{D}}(n_1,\dots ,n_k)={\delta }_{\mathrm{m},\mathcal{D}}^{-}(n_1,\dots ,n_k)\le {\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)={\delta }_{\mathcal{D}}(n_1,\dots ,n_k)$.

Proof. 

This is similar to the proof of Proposition 1. □

Remark 4.

The theory of ${\delta }_{\mathcal{D}}$-invariants (18) of a sub-Riemannian manifold can be developed similarly to the theory of Chen’s δ-invariants of a Riemannian manifold, and then applied to isometric immersions $f:(M,g;\mathcal{D})\to (\overline{M},\overline{g})$. For example, analogously to (15), for each k-tuple $(n_1,\dots ,n_k)\in S\left(d\right)$, we obtain the inequality ${\delta }_{\mathcal{D}}(n_1,\dots ,n_k)\le \frac{d^2(d+k-1-{\sum }_i{n}_i)}{2(d+k-{\sum }_i{n}_i)}{\parallel H_{\mathcal{D}}\parallel }^2+\frac{1}{2}[d(d-1)-{\sum }_i{n}_i(n_i-1)]max\overline{K}$. The case of equality is of special interest. Corresponding extremal immersions in Euclidean space in terms of ${\delta }_{\mathcal{D}}$-invariants are the sub-Riemannian analogue of Chen’s “ideal immersions".

2. Next, we consider the case when a distribution $\mathcal{D}$ is represented as the sum of $k\ge 2$ mutually orthogonal distributions of ranks $n_i>0$: $\mathcal{D}={\mathcal{D}}_1\oplus \dots \oplus {\mathcal{D}}_k$; see [12]. Thus, ${\sum }_{i=1}^k{n}_i=d$. Let $x\in M$ and $\left\{e_i\right\}$ on $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ be an adapted orthonormal frame on $\mathcal{D}\left(x\right)$, i.e.,

$$\{e_1,\dots ,e_{n_1}\}\subset {\mathcal{D}}_1\left(x\right),\dots \{e_{n_{k-1}+1},\dots ,e_{n_k}\}\subset {\mathcal{D}}_k\left(x\right).$$

The mutual curvature of $({\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ is a function on M defined by (1) with $V_i={\mathcal{D}}_i\left(x\right)$,

$${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)={\sum }_{i<j}{\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_i,{\mathcal{D}}_j).$$

Here, ${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_i,{\mathcal{D}}_j)$ is the mutual curvature of the pair $({\mathcal{D}}_i,{\mathcal{D}}_j)$ given at $x\in M$ by

$${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_i\left(x\right),{\mathcal{D}}_j\left(x\right))={\sum }_{n_{i-1}<a\le n_i,n_{j-1}<b\le n_j}K(e_a,e_b),i\ne j,$$

and does not depend on the choice of frames.

Remark 5.

If $\mathcal{D}=TM$, i.e., $d=n$, then we obtain an almost k-product manifold $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ (e.g., a multiply twisted or warped product). In this case, ${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ is the mixed scalar curvature; see [3].

An isometric immersion $f:(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)\to (\overline{M},\overline{g})$ is called mixed totally geodesic on $\mathcal{D}$ if

$$h(X,Y)=0\mathrm{for}\mathrm{all}X\in {\mathcal{D}}_i,Y\in {\mathcal{D}}_j,i\ne j.$$

The following result generalizes [9] (Theorem 1), where $d=n$.

Theorem 4.

Let $f:(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)\to (\overline{M},\overline{g})$ be an isometric immersion, and $\mathcal{D}={⨁}_{i=1}^k{\mathcal{D}}_i$. Then,

$${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)\le \frac{k-1}{2k}{\parallel H_{\mathcal{D}}\parallel }^2+{\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k).$$

(19)

The equality in (19) holds at a point $x\in M$ if and only if f is mixed totally geodesic on ${\mathcal{D}}_x$, $H_1\left(x\right)=\dots =H_k\left(x\right)$ and ${\overline{\mathrm{S}}}_{\mathrm{m}}({\mathcal{D}}_1\left(x\right)$, $\dots ,{\mathcal{D}}_k\left(x\right))={\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)\left(x\right)$.

Proof. 

The proof of the first claim is similar to the proof of Theorem 1. We take $V_i={\mathcal{D}}_i\left(x\right)$. The proof of the second assertion follows directly from the cases of equality, as in the proof of Theorem 1. □

Example 2

(see [9]). Consider distributions ${\mathcal{D}}_i(i=1,\dots ,k)$ on a domain M on a unit sphere $S^n\left(1\right)$ in ${\mathbb{R}}^{n+1}$; thus, ${\overline{\delta }}_{\mathrm{m}}^{+}(n_1,\dots ,n_k)=0$. Using coordinate charts, we can take integrable distributions ${\mathcal{D}}_i$, and M diffeomorphic to the product of k manifolds. Let, for simplicity, $\mathcal{D}=TM$.

1. For $k=2$, suppose that $M\subset S^n\left(1\right)$ is locally diffeomorphic to the product ${\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}$.

Let $n=3$ and $n_1=1,n_2=2$; then, ${\parallel H\parallel }^2=9$ and ${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,{\mathcal{D}}_2)=2$. Hence, (19) reduces to the inequality $2<9/4$ (note that $H_1=\frac{1}{3}H\ne \frac{2}{3}H=H_2$).

Let $n=4$, $n_1=n_2=2$ and locally $M\subset S^4\left(1\right)$ be diffeomorphic to ${\mathbb{R}}^2\times {\mathbb{R}}^2$. Then, ${\parallel H\parallel }^2=16$, $H_1=H_2$, ${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,{\mathcal{D}}_2)=4$ and (19) reduces to the equality $4=16/4$.

2. Let $k=3$ and $n_1=n_2=n_3=1$, i.e., we consider three one-dimensional distributions ${\mathcal{D}}_i(i=1,2,3)$ on a domain $M\subset S^3\left(1\right)\subset {\mathbb{R}}^4$. Then, ${\parallel H\parallel }^2=9$, $H_1=H_2=H_3$, ${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,{\mathcal{D}}_2,{\mathcal{D}}_3)=3$, and (19) reduces to the equality $3=(2/6)·9$.

Corollary 6.

A sub-Riemannian manifold $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ with ${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)>0$ and $\mathcal{D}={⨁}_{i=1}^k{\mathcal{D}}_i$ does not admit $\mathcal{D}$-minimal isometric immersions into Euclidean space ${\mathbb{R}}^m$.

Let $max{\overline{\tau }}_k$ be the maximum of scalar curvature over all k-dimensional subspaces of $\overline{M}$.

Corollary 7.

Let $f:(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)\to (\overline{M},\overline{g})$ be an isometric immersion, $\mathcal{D}={⨁}_{i=1}^k{\mathcal{D}}_i$, and all distributions ${\mathcal{D}}_i$ be one-dimensional, i.e., $k=d$. Then, we obtain the inequality

$$\tau \left(\mathcal{D}\right)\le \frac{k-1}{2k}{\parallel H_{\mathcal{D}}\parallel }^2+max{\overline{\tau }}_k.$$

(20)

The equality in (20) holds at a point $x\in M$ if and only if f is totally umbilical on ${\mathcal{D}}_x$, i.e., $h_{|{\mathcal{D}}_x}=H_{{\mathcal{D}}_x}·g_{|{\mathcal{D}}_x}$, and $\tau \left({\mathcal{D}}_x\right)=max{\overline{\tau }}_k\left(x\right)$.

Proof. 

This follows directly from (19) with $n_i=1$. □

3. Finally, consider so-called adapted isometric immersions $f:(M,g;\mathcal{D})\to (\overline{M},\overline{g};\overline{\mathcal{D}})$ of sub-Riemannian manifolds, i.e., $f_{*}\left(\mathcal{D}\right)\subset {\overline{\mathcal{D}}}_{\left|f\right(M)}$. Such a structure on a smooth manifold M can be obtained from a special immersion of M in $(\overline{M},\overline{g};\overline{\mathcal{D}})$. Let $f_{*}\left(TM\right)$ intersect transversally with the distribution $\overline{\mathcal{D}}$ restricted to $f\left(M\right)$; then, $f:M\to \overline{M}$ induces a required distribution $\mathcal{D}=f_{*}^{-1}(\overline{\mathcal{D}}\cap f\left(TM\right))$ on M with induced metric g. If the distributions $\mathcal{D}$ and $\overline{\mathcal{D}}$ are represented as the sums of $k\ge 2$ mutually orthogonal distributions, i.e., $\mathcal{D}={\mathcal{D}}_1\oplus \dots \oplus {\mathcal{D}}_k$ and $\overline{\mathcal{D}}={\overline{\mathcal{D}}}_1\oplus \dots \oplus {\overline{\mathcal{D}}}_k$, then we also assume the following condition: $f_{*}\left({\mathcal{D}}_i\right)\subset {\overline{\mathcal{D}}}_{i\left|f\right(M)}$ for $i=1,\dots ,k$.

Theorem 5.

Let $f:(M,g;\mathcal{D})\to (\overline{M},\overline{g};\overline{\mathcal{D}})$ be an adapted isometric immersion and ${\sum }_i{n}_i=s\le d$. Then,

$${\delta }_{\mathrm{m},\mathcal{D}}^{+}(n_1,\dots ,n_k)\le {\overline{\delta }}_{\mathrm{m},\overline{\mathcal{D}}}^{+}(n_1,\dots ,n_k)+\frac{k-1}{2k}\left\{\begin{array}{cc}{\mathcal{H}}_{\mathcal{D}}{\left(s\right)}^2,& \mathrm{if}s<d,\\ \parallel H_{\mathcal{D}}{\parallel }^2,& \mathrm{if}s=d.\end{array}\right.$$

(21)

The equality in (21) holds at a point $x\in M$ if and only if there exist mutually orthogonal subspaces $V_1,\dots ,V_k$ of ${\mathcal{D}}_x$ such that f is mixed totally geodesic on $V={⨁}_{i=1}^k{V}_i$, $H_1\left(x\right)=\dots =H_k\left(x\right)$ and ${\overline{\mathrm{S}}}_{\mathrm{m}}(V_1,\dots ,V_k)={\overline{\delta }}_{\mathrm{m},{\overline{\mathcal{D}}}_x}^{+}(n_1,\dots ,n_k)$.

Proof. 

This is similar to the proof of Theorem 3. □

Theorem 6.

Let $f:(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)\to (\overline{M},\overline{g};{\overline{\mathcal{D}}}_1,\dots ,{\overline{\mathcal{D}}}_k)$ be an adapted isometric immersion and ${\sum }_i{n}_i=d$. Then,

$${\mathrm{S}}_{\mathrm{m}}({\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)\le \frac{k-1}{2k}{\parallel H_{\mathcal{D}}\parallel }^2+{\overline{\delta }}_{\mathrm{m},\overline{\mathcal{D}}}^{+}(n_1,\dots ,n_k).$$

(22)

where $\mathcal{D}={⨁}_{i=1}^k{\mathcal{D}}_i$ and $\overline{\mathcal{D}}={⨁}_{i=1}^k{\overline{\mathcal{D}}}_i$. The equality in (22) holds at a point $x\in M$ if and only if f is mixed totally geodesic on ${\mathcal{D}}_x$, $H_1\left(x\right)=\dots =H_k\left(x\right)$ and ${\overline{\mathrm{S}}}_{\mathrm{m}}({\mathcal{D}}_1\left(x\right)$, $\dots ,{\mathcal{D}}_k\left(x\right))={\overline{\delta }}_{\mathrm{m},{\overline{\mathcal{D}}}_x}^{+}(n_1,\dots ,n_k)$.

Proof. 

The proof of the first claim is similar to the proof of Theorem 1. We take $V_i={\mathcal{D}}_i\left(x\right)$ at each point $x\in M$. The proof of the second assertion follows from the cases of equality in the proof of Theorem 1. □

5. Conclusions

We supplemented the classical problem with the question of finding a simple optimal connection between the intrinsic and extrinsic invariants of a submanifold equipped with mutually orthogonal distributions or foliations, or of a sub-Riemannian manifold, isometrically immersed into another sub-Riemannian manifold in an adapted way. The main contribution of the paper is the concept of ${\delta }_{\mathrm{m}}$-invariants of a Riemannian manifold, based on the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. Similarly to the ${\delta }_{\mathrm{m}}$-invariants (which are different from Chen invariants), we introduced ${\delta }_{\mathrm{m},\mathcal{D}}$-invariants and also Chen-type ${\delta }_{\mathcal{D}}$-invariants for a sub-Riemannian manifold $(M,g;\mathcal{D})$. The ${\delta }_{\mathrm{m}}$-, ${\delta }_{\mathcal{D}}$- and ${\delta }_{\mathrm{m},\mathcal{D}}$-invariants were compared with Chen’s $\delta$-invariants, but a deeper study of such relationships is needed. We used these invariants to prove new geometric inequalities involving the squared intermediate mean curvature for a (sub-)Riemannian submanifold and for a submanifold equipped with mutually orthogonal noncomplementary distributions. Some consequences of the absence of minimal isometric immersions in a Euclidean space were given. In particular, the main results of the paper are six theorems, four propositions and seven corollaries. We delegate to the future a deeper study of the case of equality in the obtained geometric inequalities, the connection with Chen’s “ideal immersions", and the search for applications of ${\delta }_{\mathrm{m}}$-, ${\delta }_{\mathcal{D}}$- and ${\delta }_{\mathcal{D}}$-invariants in the presence of (para)-Kähler, contact or affine structures and submersions.