Integral Formulas for Almost Product Manifolds and Foliations

Abstract

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with $k=2,3$ distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds.

1. Introduction

Distributions, i.e., sub-bundles of the tangent bundle on a manifold, arise in differential geometries as line fields, submersions, fiber bundles, Lie groups actions, almost product manifolds and in theoretical physics. A pair of complementary orthogonal distributions on a Riemannian manifold is called an almost product structure; see [1]. Geometers and engineers are also interested in singular distributions, i.e., having varying dimensions; see [2]. Foliations, being partitions of a manifold into collections of submanifolds (leaves), generate integrable distributions. The extrinsic geometry of foliated Riemannian manifolds is responsible for the properties related to the second fundamental form of the leaves and its invariants (e.g., the principal curvatures). The extrinsic geometry of foliations includes the following topics [3]:

• Integral formulas, which provide geometrical obstructions for the existence of foliations, with the given geometry of the leaves being totally geodesic, totally umbilical or minimal and, in extreme cases, leading to splitting results.

• Variational formulas for geometric quantities by a change in metric, which are used in exploring extrinsic geometric flows and actions on foliations.

• Prescribing the extrinsic geometry of foliations by an adapted (e.g., biconformal) change in metric.

A Riemannian almost multi-product manifold is a Riemannian manifold equipped with $k\ge 2$ pairwise orthogonal complementary distributions ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$. We meet this structure in such topics of differential geometry as multiply-warped (or twisted) products and the webs composed of foliations; see [4,5]. Let $M_2=F_2\times \dots \times F_k$ be the product of $k-1$ manifolds $F_i$, and let $u=(u_2,\dots ,u_k)$. A multiply twisted product $F_1{\times }_u{M}_2$ is the product manifold $F_1\times M_2$ with the metric $g=g_{F_1}\oplus u_2^2{g}_{F_2}\oplus \dots \oplus u_k^2{g}_{F_k}$, where $u_i:F_1\times F_i\to (0,\infty )$ for each $i\ge 2$. This can be extended as a multiply doubly twisted product $F_1{\times }_{(u_1,u)}{M}_2$ with $u_1:F_1\times M_2\to (0,\infty )$, where its leaves $F_1\times \left\{y\right\}$ and fibers corresponding to $F_i(i\ge 2)$ are totally umbilical submanifolds. For a multiply twisted product, i.e., $u_1=1$, the leaves are even totally geodesic. For $k=2$, we obtained a doubly twisted product $F_1{\times }_{(u_1,u_2)}{F}_2$ of Riemannian manifolds $(F_i,g_i)$ with positive warping functions $u_i\in {\mathrm{C}}^{\infty }(F_1\times F_2)$; see [6]. In this case, the second fundamental tensors $h_i$ and the mean curvature vector fields $H_i$ of the distributions are given (using orthoprojectors $P_i:TM\to T{F}_i$ by

$h_1=-P_2\nabla \left(\log u_1\right)g_1,h_2=-P_1\nabla \left(\log u_2\right)g_2,H_1=-n_1{P}_2\nabla \left(\log u_1\right),H_2=-n_2{P}_1\nabla \left(\log u_2\right),$

where ∇ is the Levi–Civita connection of $(M,g)$. One can ask when $(M,g,{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ splits locally into the product of k manifolds. The best known answer to this question is the de Rham’s decomposition theorem, e.g., [7] if each distribution ${\mathcal{D}}_i$ is parallel (with respect to ∇), then M splits, i.e., each point $x\in M$ has a neighborhood U, which is a product $F_1\times \dots \times F_k$ of Riemannian manifolds such that the submanifolds, which are parallel to the factor $F_i$, are integral manifolds of the distribution ${\mathcal{D}}_i{|}_U$. If M is complete and simply connected, the assertion is true with $U=M$. This theorem was applied to multiply twisted products and to more general foliations.

This review article presents an overview of integral formulas for foliations and multiply twisted products, as well as for Riemannian and metric-affine almost multi-product manifolds. Under some conditions, the integral formulas provide splitting results and geometrical obstructions for the existence of foliations. To simplify the presentation of the results, we do not consider pseudo-Riemannian metrics, but restrict ourselves to only Riemannian metrics. We also omit results related to integral formulas along a leaf of a foliation and integral formulas for codimension one foliated Finsler spaces.

2. Riemannian Almost Product Structure

In Section 2.1, we start from the result by G. Reeb [8], and, after that, we consider a series of integral formulas for a codimension one foliation, including the case of an ambient space of constant curvature; see [9,10,11,12,13]. One can ask the question: can these results be generalized for a foliation of an arbitrary codimension? In Section 2.2, we represent a series of integral formulas for a foliation of any codimension or a Riemannian almost product structure—see [14,15]—that starts from the result by P. Walczak [16] (see also [12,17,18]) and includes the case of an ambient space of constant curvature; see [19]. These formulas include functions $f_j(0\le j<n)$ depending on the scalar invariants of the shape operator of the leaves; see [3]. These reduce to formulas with Newton transformations of the shape operator for a special choice of functions $f_j$. In Section 2.3, we discuss integral formulas for a metric-affine space equipped with two complementary orthogonal distributions; see [3,20]. The integrand includes the mixed scalar curvature and invariants of the second fundamental forms of the foliation. Under some conditions, this provides splitting results and geometrical obstructions for the existence of foliations. In Section 2.4, we consider singular distributions as images of the tangent bundle under smooth endomorphisms; see [21]. Using a modified divergence theorem for a pair of transverse singular distributions gives an equality with a modified mixed scalar curvature and a modified divergence of a geometrically interesting vector field. This provides an integral formula for two singular distributions.

2.1. Codimension One Foliations

Let a Riemannian manifold $(M^{n+1},g)$ be equipped with a codimension one foliation $\mathcal{F}$, and let N be a unit normal to the leaves of $\mathcal{F}$. The shape (Weingarten) operator $A_N:T\mathcal{F}\to T\mathcal{F}$ of the leaves is given by $A_N\left(X\right)=-{\nabla }_{X}N$, where ∇ is the Levi–Civita connection. The generalized mean curvatures ${\sigma }_r={\sigma }_r\left(A_N\right)$ are functions on M defined as coefficients of the n-th degree polynomial $\det (\mathrm{id}+t{A}_N)$ in t, i.e.,

(1)$\det (\mathrm{id}+t{A}_N)={\sum }_{r=0}^n{\sigma }_r\left(A_N\right)t^r,$

where $\mathrm{id}:T\mathcal{F}\to T\mathcal{F}$ is the identity map. Thus, ${\sigma }_0=\det \mathrm{id}=1,{\sigma }_1=\mathrm{trace}{A}_N,\dots ,{\sigma }_n=\det A_N$.

The first known integral formula for a foliation of codimension one is due to Reeb [8]:

(2)${\int }_{M}Hd{\mathrm{vol}}_g=0,\mathrm{where}H={\sigma }_1\mathrm{is}\mathrm{the}\mathrm{mean}\mathrm{curvature}\mathrm{of}\mathrm{the}\mathrm{leaves};$

thus, either $H\equiv 0$ or $H\left(x\right)H\left(x^{\prime }\right)<0$ for some points $x\ne x^{\prime }$ on M. Its proof is based on the divergence theorem, and the identity $\mathrm{div}N=-H$. Note that $H={\sigma }_1=\mathrm{trace}{A}_N$.

The second formula in the series of integrated ${\sigma }_i$s states that the integrated ${\sigma }_2$ is half of the integrated Ricci curvature in the N-direction; see [11]:

(3)${\int }_M({\sigma }_2-\frac{1}{2}{\mathrm{Ric}}_{N,N})d{\mathrm{vol}}_g=0,$

which is a consequence of the divergence theorem applied to ${\nabla }_{N}N+{\sigma }_{1}N$. Here, ${\nabla }_{N}N$ is the curvature vector of the curves orthogonal to the leaves of $\mathcal{F}$ and ${\mathrm{Ric}}_{N,N}=\mathrm{trace}(X\to R_{X,N}N)$, where $R_{X,Y}={\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$ is the curvature tensor of $(M,g)$.

For $n=1$, Formula (3) reduces to the integral of the Gaussian curvature, ${\int }_{M}Kd{\mathrm{vol}}_g=0$.

Example 1.

Let a compact Riemannian manifold $(M,g)$ with $\mathrm{Ric}\ge 2c>0$ be endowed with a codimension one foliation. Using (2), we obtain ${\sigma }_1\left(x\right)=0$ at some $x\in M$. Then, ${\sigma }_2\left(x\right)=-{\sum }_i{k}_i^2\left(x\right)\le 0$. Using (3), ${\int }_M{\sigma }_{2}d{\mathrm{vol}}_g\ge c\mathrm{Vol}(M,g)$. Thus, ${\sigma }_2\left(y\right)>c$ at some point $y\in M$ (otherwise, ${\sigma }_2\le c$ on M, hence ${\sigma }_2\equiv c>0$ on M—a contradiction to the property ${\sigma }_2\left(x\right)\le 0$). We conclude that the image of the function ${\sigma }_2:M\to \mathbb{R}$ contains the whole interval $[0,c+\epsilon ]$ for some $\epsilon >0$.

Brito, Langevin and Rosenberg [10] (following the result by Asimov [22]) proved that the integrals of ${\sigma }_r$ of a codimension one foliation on a compact manifold $M^{n+1}\left(c\right)$ of constant curvature c do not depend on the foliation: they depend on r, n, c and the volume of $(M,g)$ only:

(4)${\int }_M{\sigma }_{r}d{\mathrm{vol}}_g=\left\{\begin{array}{cc}{c}^{r/2}\left(\begin{array}{c}n/2\\ r/2\end{array}\right)\mathrm{Vol}(M,g),& n\mathrm{and}r\mathrm{even},\\ 0,& \mathrm{either}n\mathrm{or}r\mathrm{odd}.\end{array}\right.$

The Newton transformations $T_r\left(A_N\right)$ of $A_N$ are defined inductively by $T_0\left(A_N\right)=\mathrm{id},T_r\left(A_N\right)={\sigma }_r\left(N\right)\mathrm{id}-A_N{T}_{r-1}$$\left(A_N\right)(1\le r\le n)$, or explicitly by $T_r\left(A_N\right)={\sigma }_r\mathrm{id}-{\sigma }_{r-1}$$A_N+\dots +{(-1)}^r{A}_N^r$. For the shape operator $A_N$, we have (for the traces of tensors on $\mathcal{F}$)

$\begin{array}{ccc}& & {\mathrm{trace}}_{\mathcal{F}}{T}_r\left(A_N\right)=(n-r){\sigma }_r,{\mathrm{trace}}_{\mathcal{F}}(A_N·T_r\left(A_N\right))=(r+1){\sigma }_{r+1},\hfill \\ & & {\mathrm{trace}}_{\mathcal{F}}(A_N^2·T_r\left(A_N\right))={\sigma }_1{\sigma }_{r+1}-(r+2){\sigma }_{r+2},\hfill \\ & & {\mathrm{trace}}_{\mathcal{F}}\left(T_{r-1}\left(A_N\right)\left({\nabla }_X^{\mathcal{F}}{A}_N\right)\right)=X\left({\sigma }_r\right),X\in T\mathcal{F},r>0.\hfill \end{array}$

One can apply the divergence theorem to a geometrically interesting vector field to represent a series of integral formulas for a foliation of codimension one of a closed Riemannian manifold. The $\mathcal{F}$-divergence of a vector field X tangent to $\mathcal{F}$ on $(M,g)$ is defined using a local orthonormal basis $\left\{e_i\right\}$ of $T\mathcal{F}$ by

${\mathrm{div}}_{\mathcal{F}}X={\mathrm{trace}}_{\mathcal{F}}(Y\to {\nabla }_{Y}X)={\sum }_{i=1}^n⟨{\nabla }_{e_i}X,e_i⟩.$

For a $(1,k)$-tensor S, we put ${\mathrm{div}}_{\mathcal{F}}S={\mathrm{trace}}_{\mathcal{F}}(Y\to {\nabla }_{Y}S)$; that is,

(5)$\left({\mathrm{div}}_{\mathcal{F}}S\right)(X_1,\dots ,X_k)={\sum }_i⟨\left({\nabla }_{e_i}S\right)(X_1,\dots ,X_k),e_i⟩.$

Consequently, the $\mathcal{F}$-divergence of the Newton transformation $T_r\left(A_N\right)$ is defined by (see [9])

${\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_N\right)={\sum }_{i=1}^n\left({\nabla }_{e_i}{T}_r\left(A_N\right)\right)e_i.$

Observe that ${\mathrm{div}}_{\mathcal{F}}{T}_0\left(A_N\right)=0$. Define a linear operator $R\left(X\right):T\mathcal{F}\to T\mathcal{F}$ (for $X\in TM$) by $R\left(X\right):Z\to {\left(R(Z,X)N\right)}^{\top }$. The following formula generalizing (4) was obtained in [9]:

(6)${\int }_M(r+2){\sigma }_{r+2}-⟨{\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_N\right),{\nabla }_{N}N⟩-\mathrm{trace}\left(T_r\left(A_N\right)R\left(N\right)\right)d{\mathrm{vol}}_g=0,$

where $⟨{\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_N\right),Z⟩={\sum }_{j=1}^r\mathrm{trace}(R({(-A_N)}^{j-1}Z)T_{r-j}(A_N))$ for any $Z\in T\mathcal{F}$.

Other integral formulas for foliations of codimension one on a Riemannian manifold of finite volume (which are nicely applicable for symmetric spaces) were obtained in [13]; they generalize (4) and start from (3). For $r=0$, (6) reads as (3), and for $r=1$, (6) reduces to

${\int }_M(3{\sigma }_3-{\sigma }_1{\mathrm{Ric}}_{N,N}+\mathrm{trace}\left(A_{N}R\left(N\right)\right)-{\mathrm{Ric}}_{N,{\nabla }_{N}N})d{\mathrm{vol}}_g=0.$

Consequently, for a totally umbilical foliation of a closed Einstein manifold (with $\mathrm{Ric}=nc·g$ for some $c\in \mathbb{R}$), (6) reduces to (4) (see review [23] on integral formulas on foliations of codimension one).

We believe that these integral formulas with ${\sigma }_i$ serve as main obstructions for recovering metrics by higher mean curvatures of codimension one foliations.

In [24], we generalized (6) for a general operator $\mathcal{A}\left(A_N\right)={\sum }_{j=0}^{n-1}{f}_j({\tau }_1,\dots ,{\tau }_n){\left(A_N\right)}^j$ with given functions $f_j$ on ${\mathbb{R}}^n(0\le j<n)$ instead of $T_r\left(A_N\right)$. The choice of $\mathcal{A}\left(A_N\right)$ is justified for the following reasons: (i) the powers ${\left(A_N\right)}^j$ are the only (1,1)-tensors, obtained algebraically from the shape operator $A_N$, while ${\tau }_r=\mathrm{trace}\left({\left(A_N\right)}^r\right)(r=1,\dots ,n)$, or, equivalently, ${\sigma }_i(1\le i\le n)$ generate all scalar invariants of $A_N$; (ii) the Newton transformation $T_r\left(A_N\right)(r<n)$ depends on all ${\left(A_N\right)}^j(j\le r)$.

We allow the singular case because there are no codimension one foliations on many manifolds, while any manifold admits such foliations on the complement of some set. An example of foliations with singularities is the “open book decompositions” of manifolds. Here, we consider a foliation $\mathcal{F}$ defined on the complement $M\backslash {\Sigma }_k$ of a union ${\Sigma }_k$ of finitely many closed codimension $k\ge 2$ submanifolds of a manifold M. (In fact, this is not a foliation of M.) Using the Stokes theorem, if $(M,g)$ is closed-oriented and a Riemannian manifold, X is a vector field on an open set $M\backslash {\Sigma }_k$, $(k-1)(p-1)\ge 1$ and ${\int }_M{\parallel X\parallel }^{p}d{\mathrm{vol}}_g<\infty$; see [25]:

(7)${\int }_M\left(\mathrm{div}X\right)d{\mathrm{vol}}_g=0.$

We give first applications of the above with $k=2$; see ([3], Section 2.2.2) and [25].

Theorem 1.

Let $\mathcal{F}$ be a codimension one foliation, defined on the complement to the singular locus ${\Sigma }_2$ of codimension $\ge 2$ on a closed Riemannian manifold $(M,g)$. Then, (2) is valid. If, in addition, ${\int }_M({\sigma }_1^2+\parallel {\nabla }_{N}N{\parallel }^2)d{\mathrm{vol}}_g<\infty$, then (3) is valid.

2.2. Foliations of Arbitrary Codimension

One can generalize (6) for a foliation of an arbitrary codimension as a series of integral formulas depending on Newton transformations related to the shape operator. The idea is to compute the divergence of certain vector fields and to apply the divergence theorem.

Let $(M,g)$ be a Riemannian manifold with the Levi–Civita connection ∇, and let $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$ be two complementary orthogonal distributions on M (i.e., smooth sub-bundles of the tangent bundle $TM$). Set $n=\dim \mathcal{D}$ and $p=\dim {\mathcal{D}}^{\perp }$. Hence, $\dim M=n+p$. The second fundamental form h and integrability tensor T of $\mathcal{D}$ (and, similarly, of ${\mathcal{D}}^{\perp }$) are defined as follows:

$\begin{array}{c}\hfill h(X,Y)=\frac{1}{2}{({\nabla }_{X}Y+{\nabla }_{Y}X)}^{\perp },T(X,Y)=\frac{1}{2}{({\nabla }_{X}Y-{\nabla }_{Y}X)}^{\perp }.\end{array}$

A distribution $\mathcal{D}$ is integrable, i.e., tangent to a foliation, if $T=0$. A distribution (or a foliation associated with integrable distribution) on a Riemannian manifold is called totally geodesic if its second fundamental tensor vanishes; in this case, any geodesic of a manifold that is tangent to the distribution at one point is tangent to it at all points. Denote $H={\mathrm{trace}}_{g}h$ and $H^{\perp }=\mathrm{trace}{h}^{\perp }$ as the mean curvature vectors of $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$, respectively. We call $\mathcal{D}$ harmonic if $H=0$ and totally umbilical if $h=(H/n)g$.

Brito and Naveira [19] have shown for a p-dimensional totally geodesic foliation (tangent to ${\mathcal{D}}^{\perp }$) on a compact space form $M^{n+p}\left(c\right)$ that integrated extrinsic mean curvatures, ${\gamma }_{2s}\left(\mathcal{D}\right)$, depend on $s,p,n,c$ and the volume of $(M,g)$ only. A similar result for the integrated ${\sigma }_{2s}\left(\mathcal{D}\right)$ was proven in [13]:

(8)${\int }_M{\sigma }_{2s}d{\mathrm{vol}}_g=\left\{\begin{array}{cc}\left(\begin{array}{c}n/2\\ s\end{array}\right)\frac{2{\pi }^{p/2}}{\Gamma (p/2)}{c}^s\mathrm{Vol}(M,g),& n\mathrm{even},\\ 0,& n\mathrm{odd}.\end{array}\right.$

Since $\mathcal{D}$ is tangent to a totally geodesic foliation, the constant c must be non-negative. When $\mathcal{D}$ is transversally orientable, (8) reduces to (4) with a doubled right hand side.

A plane, which nontrivially intersects both distributions, is called mixed, and $S_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })={\sum }_{i\le n}{\sum }_{\alpha \le p}g(R(e_i,e_{\alpha })e_i,e_{\alpha })$ is called the mixed scalar curvature, it does not depend on choices of a local orthonormal frame $\{e_i,e_{\alpha }\}$ of $TM$ adapted for $({\mathcal{D}}^{\perp },\mathcal{D})$. The following integral formula for complementary orthogonal distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$ on a compact manifold $(M,g)$ was given in [16]:

(9)${\int }_M\left(S_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })+{\parallel h\parallel }^2+{\parallel h^{\perp }\parallel }^2-{\parallel H\parallel }^2-{\parallel H^{\perp }\parallel }^2-{\parallel T\parallel }^2-{\parallel T^{\perp }\parallel }^2\right)d{\mathrm{vol}}_g=0.$

One can prove (9) by applying the divergence theorem to the identity

(10)$\mathrm{div}(H+H^{\perp })=S_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })+{\parallel h\parallel }^2+{\parallel h^{\perp }\parallel }^2-{\parallel H\parallel }^2-{\parallel H^{\perp }\parallel }^2-{\parallel T\parallel }^2-{\parallel T^{\perp }\parallel }^2.$

For a foliation of codimension one, (9) reduces to (3). Formula (9) was applied in differential geometry to harmonic morphisms and holomorphic distributions on Kähler manifolds—see [18]—and almost contact metric manifolds; see [26]. The approach of calculating the divergence of $H+H^{\perp }$ in (9) was extended in [14] to vector fields ${(A_{H^{\perp }}^{\perp })}^{k}H+{\left(A_H\right)}^k{H}^{\perp }$ for $k>0$, where $A_{H^{\perp }}^{\perp }H=-{\left({\nabla }_H{H}^{\perp }\right)}^{\perp }$. For $k=1$, we have

$\mathrm{div}(A_{H^{\perp }}^{\perp }H+A_H{H}^{\perp })={\mathrm{Ric}}_{H,H^{\perp }}+Q_1,$

where $Q_1$ has a complicated form, but, if the distributions define a totally umbilical foliation of constant mean curvature, then $Q_1=-(\frac{1}{n}+\frac{1}{p}){\parallel H\parallel }^2·{\parallel H^{\perp }\parallel }^2$.

Below, we assume, for simplicity, that ${\mathcal{D}}^{\perp }$ defines a foliation $\mathcal{F}$. Let $S^{\perp }=\{\xi \in \mathcal{D}:\parallel \xi \parallel =1\}$ be the unit sphere bundle with the Sasaki metric and the volume form $d\omega$. The natural projection $\pi :S^{\perp }\to M$ is a Riemannian submersion with totally geodesic fibers ${\left\{S_x^{\perp }\right\}}_{x\in M}$—unit spheres. Hence, $\mathrm{d}\mathrm{vol}(S^{\perp },d\omega )$ is the product of $\mathrm{d}\mathrm{vol}\left(S_1^{n-1}\right)$ and $\mathrm{d}{\mathrm{vol}}_g$, and the differentiating along M commutes with the integration along the fibers $S_x^{\perp }$. Let ${\tau }_r\left(\xi \right)=\mathrm{trace}\left({\left(A_{\xi }\right)}^r\right)$ be the power sums symmetric functions. Given functions $f_j$ on ${\mathbb{R}}^p(0\le j<p)$, define a (1,1)-tensor field $\mathcal{A}$ by

$\mathcal{A}={\int }_{\xi \in S_x^{\perp }}{\mathcal{A}}_{\xi }\mathrm{d}\omega ,\mathrm{where}{\mathcal{A}}_{\xi }:={\sum }_{j=0}^{p-1}{f}_j({\tau }_1\left(\xi \right),\dots ,{\tau }_p\left(\xi \right)){\left(A_{\xi }\right)}^j\mathrm{and}x\in M.$

Evidently, $⟨A_{\xi }\left(X\right),Y⟩=⟨h_{X,Y},\xi ⟩$ for $X,Y\in {\mathcal{D}}^{\perp }$. Define a linear operator $R(X,Y):{\mathcal{D}}^{\perp }\to {\mathcal{D}}^{\perp }$ by

$R(X,Y):Z\to {\left(R(Z,X)Y\right)}^{\top },Z\in {\mathcal{D}}^{\perp },X,Y\in TM.$

Set $R\left(\xi \right)=R(\xi ,\xi )$. Let $\left\{e_i\right\}$ be a local orthonormal frame of $T\mathcal{F}$.

Lemma 1

(see [15]). The $\mathcal{F}$-divergence of ${\left(A_{\xi }\right)}^k$ for $k>0$—see definition in (5)—is given by the formula

${\mathrm{div}}_{\mathcal{F}}{(A_{\xi })}^k={\sum }_{1\le j\le k}\left({(k-j+1)}^{-1}{(A_{\xi })}^{j-1}{(\nabla {\tau }_{k-j+1}(A_{\xi }))}^{\top }-{\sum }_i{(A_{\xi })}^{j-1}{(R(\xi ,{(A_{\xi })}^{k-j}{e}_i)e_i)}^{\top }\right).$

Using Lemma 1 with $f_j={(-1)}^j{\sigma }_{r-j}\left(A_{\xi }\right)$ and $j\le r$, we have, for Newton transformations of $A_{\xi }$,

(11)${\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_{\xi }\right)={\sum }_{1\le j\le r}{\left({\sum }_i{(-A_{\xi })}^{j-1}\left(R(\xi ,T_{r-j}\left(A_{\xi }\right)e_i)e_i\right)\right)}^{\top }.$

Of course, using an inductive definition (see Section 2.1), we obtain

${\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_{\xi }\right)={(\nabla {\sigma }_r\left(A_{\xi }\right))}^{\top }-A_{\xi }{\mathrm{div}}_{\mathcal{F}}{T}_{r-1}\left(A_{\xi }\right)-{\sum }_i{\left({\nabla }_{e_i}A\right)}_{\xi }{T}_{r-1}\left(A_{\xi }\right)e_i.$

Using the identity $X\left({\sigma }_r\right)\left(A_{\xi }\right)=\mathrm{trace}(T_{r-1}\left(A_{\xi }\right){\left({\nabla }_{X}A\right)}_{\xi })$ for $X\in T\mathcal{F}$, and the Codazzi equation gives

${\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_{\xi }\right)=-A_{\xi }{\mathrm{div}}_{\mathcal{F}}{T}_{r-1}\left(A_{\xi }\right)+{\sum }_i{\left(R(\xi ,T_{r-1}\left(A_{\xi }\right)e_i)e_i\right)}^{\top }.$

By induction, one may show that ${\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_{\xi }\right)$ for $r>0$ is given by (11).

Lemma 2

(see [15]). There exists a local orthonormal frame $\{e_i,e_{\alpha }\}$ of $TM$ adapted for $({\mathcal{D}}^{\perp },\mathcal{D})$ such that ${\left({\nabla }_X{e}_i\left(x\right)\right)}^{\top }=0$ and ${\left({\nabla }_X{e}_{\alpha }\left(x\right)\right)}^{\perp }=0$ for any vector $X\in T_{x}M$. Using this frame, for any unit vector $\xi ={\sum }_{\alpha \le n}{y}_{\alpha }{e}_{\alpha }\in S_x^{\perp }$ with $y_{\alpha }\in \mathbb{R}$, we have, at the point $x\in M$,

$⟨{\nabla }_{e_i}{\nabla }_{\xi }\xi ,e_j⟩={({\left(A_{\xi }\right)}^2)}_{ij}+⟨R(e_i,\xi )\xi ,e_j⟩-{\left({\nabla }_{\xi }{A}_{\xi }\right)}_{ij}+{\sum }_{\alpha \le n}⟨{\nabla }_{\xi }{e}_{\alpha },e_i⟩⟨{\nabla }_{e_{\alpha }}\xi ,e_j⟩.$

Put $Z_{\xi }={\left({\nabla }_{\xi }\xi \right)}^{\top }$ for a vector field $\xi$ in $S^{\perp }$. Applying Lemma 2 and $f_j={(-1)}^j{\sigma }_{r-j}$$\left(A_{\xi }\right)(j\le r)$ and (11), we obtain that, for any $x\in M$,

$\begin{array}{c}{\mathrm{div}}_{\mathcal{F}}({\int }_{S_x^{\perp }}{T}_r(A_{\xi })Z_{\xi }\mathrm{d}\omega )={\int }_{S_x^{\perp }}(⟨-{\nabla }_{\mathcal{F}}^{*}{T}_r(A_{\xi }),Z_{\xi }⟩-\xi ({\sigma }_{r+1})(A_{\xi })-(r+2){\sigma }_{r+2}(A_{\xi })\\ +{\sigma }_1(A_{\xi }){\sigma }_{r+1}(A_{\xi })+\mathrm{trace}(T_r(A_{\xi })R(\xi ))+{\sum }_{\alpha \le n}⟨T_r(A_{\xi })({({\nabla }_{e_{\alpha }}\xi )}^{\top }),{\nabla }_{\xi }{e}_{\alpha }⟩)\mathrm{d}\omega ,\end{array}$

where $⟨{\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_{\xi }\right),Z_{\xi }⟩={\sum }_{1\le j\le r}\mathrm{trace}(T_{r-j}\left(A_{\xi }\right)R({(-A_{\xi })}^{j-1}{Z}_{\xi },\xi )$.

Theorem 2

(see [15]). On any closed Riemannian manifold $(M,g)$ with ${\mathcal{D}}^{\perp }$ tangent to a foliation $\mathcal{F}$, we have

(12)$\begin{array}{l}{\int }_{S^{\perp }}(⟨{\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_{\xi }\right),Z_{\xi }⟩-(r+2){\sigma }_{r+2}\left(A_{\xi }\right)-⟨T_r\left(A_{\xi }\right)Z_{\xi },H^{\perp }⟩\hfill \\ +\mathrm{trace}\left(T_r\left(A_{\xi }\right)R\left(\xi \right)\right)+{\sum }_{\alpha \le n}⟨T_r\left(A_{\xi }\right)\left({\left({\nabla }_{e_{\alpha }}\xi \right)}^{\top }\right),{\nabla }_{\xi }{e}_{\alpha }⟩)\mathrm{d}\omega =0.\hfill \end{array}$

We look at the first members of a series of (12). For $r=0$, it is easy to obtain

${\int }_{S^{\perp }}(-2{\sigma }_2\left(A_{\xi }\right)+\mathrm{trace}R\left(\xi \right)-⟨Z_{\xi },H^{\perp }⟩+{\sum }_{\alpha \le n}⟨{\left({\nabla }_{e_{\alpha }}\xi \right)}^{\top },{\nabla }_{\xi }{e}_{\alpha }⟩)\mathrm{d}\omega =0.$

For $r=0$, one may reduce this formula to (9), and, for $r=2$, (12) gives us (see [15]):

(13)$\begin{array}{l}{\int }_{S^{\perp }}(-4{\sigma }_4(A_{\xi })-⟨T_2(A_{\xi })Z_{\xi },H^{\perp }⟩+\mathrm{trace}(T_2(A_{\xi })R(\xi )+R(A_{\xi }{Z}_{\xi },\xi )\hfill \\ -T_1(A_{\xi })R(Z_{\xi },\xi ))+{\sum }_{\alpha \le n}⟨T_2(A_{\xi })({({\nabla }_{e_{\alpha }}\xi )}^{\top }),{\nabla }_{\xi }{e}_{\alpha }⟩)\mathrm{d}\omega =0.\hfill \end{array}$

If ${\mathcal{D}}^{\perp }$ defines a totally geodesic foliation, then (13) shortens to

${\int }_{S^{\perp }}\left(4{\sigma }_4\left(A_{\xi }\right)-\mathrm{trace}\left(T_2\left(A_{\xi }\right)R\left(\xi \right)\right)\right)\mathrm{d}\omega =0.$

2.3. Metric-Affine Manifolds

The metric-affine geometry (introduced by E. Cartan) is a generalization of Riemannian geometry. It deals with a linear connection with torsion $\overline{\nabla }$, and appears in such topics as homogeneous and almost Hermitian manifolds and Finsler geometry, e.g., [3,27]. The distinguished cases are: Riemann–Cartan manifolds, where metric-compatible connections (i.e., $\overline{\nabla }g=0$) are used, and statistical manifolds, where the connection $\overline{\nabla }$ is torsionless and the tensor $\overline{\nabla }g$ is symmetric in its entries. The main concept of information geometry is that of statistical manifold; affine hypersurfaces in ${\mathbb{R}}^{n+1}$ are a source of such manifolds. In addition, Riemann–Cartan spaces are used in gauge theory of gravity. The Levi–Civita connection ∇ is a reference point in affine space of linear connections on M, and the difference $\mathrm{T}:=\overline{\nabla }-\nabla$ is called the contorsion tensor. We associate with $\mathrm{T}$ the (1,2)-tensors ${\mathrm{T}}^{*}$ and $\widehat{\mathrm{T}}$, using the equalities

$⟨{\mathrm{T}}_X^{*}Y,Z⟩=⟨{\mathrm{T}}_{X}Z,Y⟩,{\widehat{\mathrm{T}}}_{X}Y={\mathrm{T}}_{Y}X,X,Y,Z\in {\mathrm{X}}_M.$

Note that, generally, ${(\widehat{\mathrm{T}})}^{*}\ne \widehat{{\mathrm{T}}^{*}}$. Indeed,

$\begin{array}{c}\hfill ⟨{(\widehat{\mathrm{T}})}_X^{*}Y,Z⟩=⟨{\widehat{\mathrm{T}}}_{X}Z,Y⟩=⟨{\mathrm{T}}_{Z}X,Y⟩,⟨{(\widehat{{\mathrm{T}}^{*}})}_{X}Y,Z⟩=⟨{\mathrm{T}}_Y^{*}X,Z⟩=⟨{\mathrm{T}}_{Y}Z,X⟩.\end{array}$

For a metric-compatible connection, we have ${\mathrm{T}}^{*}=-\mathrm{T}$. For a statistical connection, we obtain $\widehat{\mathrm{T}}=\mathrm{T}$ and ${\mathrm{T}}^{*}=\mathrm{T}$. Comparing the curvature tensor ${\overline{R}}_{X,Y}=[{\overline{\nabla }}_Y,{\overline{\nabla }}_X]+{\overline{\nabla }}_{[X,Y]}$ of $\overline{\nabla }$ with R, one can find

(14)$\overline{R}(X,Y)-R(X,Y)={\left({\nabla }_Y\mathrm{T}\right)}_X-{\left({\nabla }_X\mathrm{T}\right)}_Y+[{\mathrm{T}}_Y,{\mathrm{T}}_X],X,Y\in {\mathrm{X}}_M.$

The tensor $\overline{R}$ has a few symmetry properties; for example, $⟨\overline{R}(X,Y)Z,U⟩=$$-⟨\overline{R}(Y,X)Z,U⟩$.

Define the “mean curvature type” vector fields $H_{\mathrm{T}}:={\sum }_a{\epsilon }_a{\mathrm{T}}_a{E}_a$ and ${\tilde{H}}_{\mathrm{T}}:={\sum }_i{\epsilon }_i{\mathrm{T}}_i{\mathcal{E}}_i$, related to contorsion tensor $\mathrm{T}$, where $\{E_a,{\mathcal{E}}_i\}$ is a local adapted orthonormal frame. For projections of these vectors, we will use notations $H_{\mathrm{T}}^{\perp }={(H_{\mathrm{T}})}^{\perp },{\tilde{H}}_{\mathrm{T}}^{\top }={({\tilde{H}}_{\mathrm{T}})}^{\top }$, etc. Using (9) and (14), we find

$\begin{array}{ccc}\hfill {\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })& =& \frac{1}{2}{\sum }_{a,i}(⟨{\left({\nabla }_i\mathrm{T}\right)}_a{E}_a,{\mathcal{E}}_i⟩-⟨{\left({\nabla }_a\mathrm{T}\right)}_i{E}_a,{\mathcal{E}}_i⟩+⟨{\left({\nabla }_a\mathrm{T}\right)}_i{\mathcal{E}}_i,E_a⟩\hfill \\ \hfill & -& ⟨{\left({\nabla }_i\mathrm{T}\right)}_a{\mathcal{E}}_i,E_a⟩+⟨[{\mathrm{T}}_i,{\mathrm{T}}_a]E_a,{\mathcal{E}}_i⟩+⟨[{\mathrm{T}}_a,{\mathrm{T}}_i]{\mathcal{E}}_i,E_a⟩)+{\mathrm{S}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp }).\hfill \end{array}$

This construction does not depend on the adapted orthonormal frame. Denote by ${⟨B,C⟩}_{|V}$ the product of tensors restricted on the vector sub-bundle $V=(\mathcal{D}\times {\mathcal{D}}^{\perp })\oplus ({\mathcal{D}}^{\perp }\times \mathcal{D})\subset TM\times TM$. Then,

(15)$\mathrm{div}(H_{\mathrm{T}-{\mathrm{T}}^{*}}^{\perp }+{\tilde{H}}_{\mathrm{T}-{\mathrm{T}}^{*}}^{\top })=2({\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })-{\mathrm{S}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp }))-2\overline{Q},$

where

$\begin{array}{ccc}\hfill 2\overline{Q}& =& ⟨H_{\mathrm{T}},{\tilde{H}}_{{\mathrm{T}}^{*}}⟩+⟨{\tilde{H}}_{\mathrm{T}},H_{{\mathrm{T}}^{*}}⟩+⟨H_{\mathrm{T}-{\mathrm{T}}^{*}}+{\tilde{H}}_{{\mathrm{T}}^{*}-\mathrm{T}},H-H^{\perp }⟩\hfill \\ & & +⟨\mathrm{T}-{\mathrm{T}}^{*}+\widehat{\mathrm{T}}-\widehat{{\mathrm{T}}^{*}},A^{\perp }-T^{\perp ♯}+A-T^{♯}⟩-{⟨{\mathrm{T}}^{*},\widehat{\mathrm{T}}⟩}_{|V}.\hfill \end{array}$

For statistical manifolds, the above formulas reduce to

${\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })-{\mathrm{S}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })-⟨H_{\mathrm{T}},{\tilde{H}}_{\mathrm{T}}⟩+(1/2){⟨\mathrm{T},\mathrm{T}⟩}_{|V}=0.$

Using (10), (15) and (7) gives the following result in [28].

Theorem 3.

Let $(M,g,\overline{\nabla })$ be a closed metric-affine space and ${\mathcal{D}}^{\perp }$ a distribution defined outside of the “singularity set" ${\Sigma }_2$. If ${\parallel \xi \parallel }_g\in {\mathrm{L}}^2(M,g)$, where $\xi =H^{\perp }+H+\frac{1}{2}{H}_{\mathrm{T}-{\mathrm{T}}^{*}}^{\perp }+\frac{1}{2}{\tilde{H}}_{\mathrm{T}-{\mathrm{T}}^{*}}^{\top }$, then

${\int }_M\left\{{\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })-{\parallel T\parallel }^2-\parallel T^{\perp }{\parallel }^2+{\parallel h\parallel }^2+\parallel h^{\perp }{\parallel }^2-{\parallel H\parallel }^2-{\parallel H^{\perp }\parallel }^2-\overline{Q}\right\}\mathrm{d}{\mathrm{vol}}_g=0.$

This allows us to obtain splitting results. To simplify the presentation of results, assume that

(16)$\begin{array}{ccc}\hfill & & {\mathrm{T}}_{X}Y=0={\mathrm{T}}_{Y}X,{\mathrm{T}}_X^{*}Y=0={\mathrm{T}}_Y^{*}X(X\in \mathcal{D},Y\in {\mathcal{D}}^{\perp }),\hfill \end{array}$

(17)$\begin{array}{ccc}\hfill & & ⟨H_{\mathrm{T}-{\mathrm{T}}^{*}},H^{\perp }⟩=0.\hfill \end{array}$

For example, (16) provides $H_{\mathrm{T}}^{\perp }={\tilde{H}}_{\mathrm{T}}^{\top }=0$; moreover, (16) and (17) provide the vanishing of $\overline{Q}$.

We used the following version of Stokes’ theorem on a complete open Riemannian manifold.

Lemma 3

(see Proposition 1 in [29]). Let $(M,g)$ be a complete open Riemannian manifold equipped with a vector field ξ such that $\mathrm{div}\xi \ge 0$. If ${\parallel \xi \parallel }_g\in L^1(M,g)$; then, $\mathrm{div}\xi \equiv 0$.

The following theorem deals with harmonic distributions of ${\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })\ge 0$.

Theorem 4

(see [28]). Let a complete open metric-affine space $(M,g,\overline{\nabla })$ be equipped with complementary orthogonal harmonic foliations, and let conditions (16) and ${\parallel \xi \parallel }_g\in {\mathrm{L}}^1(M,g)$ with $2\xi =H_{\mathrm{T}-{\mathrm{T}}^{*}}^{\perp }+{\tilde{H}}_{\mathrm{T}-{\mathrm{T}}^{*}}^{\top }$, be satisfied. If ${\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })\ge 0$, then M splits.

Next, we consider totally umbilical distributions with ${\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })\le 0$.

Theorem 5

(see [28]). Let a complete open metric-affine space $(M,g,\overline{\nabla })$ be equipped with complementary orthogonal totally umbilical distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$. Suppose that conditions (16)

(18)$H_{\mathrm{T}}=0={\tilde{H}}_{\mathrm{T}},H_{{\mathrm{T}}^{*}}=0={\tilde{H}}_{\mathrm{T}{x}^{*}}$

and ${\parallel \xi \parallel }_g\in {\mathrm{L}}^1(M,g)$, where $\xi =H^{\perp }+H$, are satisfied. If ${\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })\le 0$, then M splits.

Totally umbilical integrable distributions appear on doubly twisted products $B{\times }_{(v,u)}F$; see the definition in Section 3. In this case, conditions (16) are obviously satisfied.

Corollary 1

(see [28]). Let the vector field $\xi =H^{\perp }+H$ on a doubly twisted product $M=B{\times }_{(v,u)}F$, which is a complete open manifold, satisfy conditions (18) and ${\parallel \xi \parallel }_g\in {\mathrm{L}}^1(M,g)$. If ${\overline{\mathrm{S}}}_{\mathrm{mix}}(\mathcal{D},{\mathcal{D}}^{\perp })\le 0$, then M is the metric product.

2.4. Singular Distributions

Singular distributions, i.e., having varying dimensions, are important for geometers and engineers, e.g., [2]. By definition, a singular distribution $\mathcal{D}$ on a manifold M assigns to each point $x\in M$ a linear subspace ${\mathcal{D}}_x\subset T_{x}M$ such that, for any $v\in {\mathcal{D}}_x$, there exists a smooth vector field V in a neighborhood U of x with the property $V\left(x\right)=v$ and $V\left(y\right)\in {\mathcal{D}}_y$ for all y of U, e.g., [2]. (This means that, locally, a singular distribution can be parameterized.) The dimension $\dim {\mathcal{D}}_x$ depends on $x\in M$. If $\dim {\mathcal{D}}_x=\mathrm{const}$, then the distribution is regular. Singular foliations can be defined as families of maximal integral submanifolds (leaves) of integrable singular distributions; hence, regular foliations correspond to integrable regular distributions.

Such distributions can be obtained as images of the tangent bundle with smooth endomorphisms. Let M be a smooth n-dimensional manifold, $TM$ be the tangent bundle, ${\mathcal{X}}_M$ be the Lie algebra of smooth vector fields on M and $\mathrm{End}\left(TM\right)$ be smooth endomorphisms of $TM$. The image $\mathcal{D}=P\left(TM\right)$ of the endomorphism $P\in \mathrm{End}\left(TM\right)$ is the singular distribution on M; see [21]. Let $P\left({\mathcal{X}}_M\right)$ be an $C^{\infty }\left(M\right)$-submodule of ${\mathcal{X}}_{\mathcal{D}}$ (smooth vector fields on $\mathcal{D}$), i.e., vector fields $P\left(X\right)\in {\mathcal{X}}_{\mathcal{D}}$, where $X\in {\mathcal{X}}_M$.

Let $P_1,P_2$ be endomorphisms of $TM$ such that $P_1\left(TM\right)\cap P_2\left(TM\right)$ is equal to zero; therefore $\mathrm{rank}{P}_1\left(x\right)+\mathrm{rank}{P}_2\left(x\right)\le n$ for all $x\in M$. For example, $P_i$ can be projectors from $TM$ onto regular distributions. Put $\mathcal{D}=P\left(TM\right)$ for $P=P_1+P_2\in \mathrm{End}\left(TM\right)$, and define distributions ${\mathcal{D}}_i=P_i\left(TM\right)$. Then, $P\left(TM\right)$ is a generalized vector sub-bundle in $TM$, but not necessarily equal to $TM$. A Riemannian metric $g=⟨·,·⟩$ on M is adapted if the following compositions vanish:

(19)$P_i{P}_j^{*}=P_i^{*}{P}_j=0(i\ne j),$

where endomorphisms $P_1^{*},P_2^{*}$ of $TM$ are adjoint, i.e., $⟨P_i\left(X\right),Y⟩=⟨X,P_i^{*}\left(Y\right)⟩$. In this case, ${\mathcal{D}}_1{\perp }_g{\mathcal{D}}_2$. The structural tensors of singular distributions ${\mathcal{D}}_i=P_i\left(TM\right)$ are tensor fields $B_i:{\left({\mathcal{X}}_M\right)}^2\to P_i\left({\mathcal{X}}_M\right)$,

$B_1(Y,X):=P_1^{*}{\nabla }_{P_{1}X}{P}_{2}Y,B_2(X,Y):=P_2^{*}{\nabla }_{P_{2}Y}{P}_{1}X.$

Similarly, define structural tensors ${\widehat{B}}_i$ of dual distributions ${\mathcal{D}}_i^{*}=P_i^{*}\left(TM\right)$ and the auxiliary tensors ${\check{B}}_i$:

$\begin{array}{ccc}& & {\widehat{B}}_1(Y,X):=P_1{\nabla }_{P_1^{*}X}{P}_2^{*}Y,{\widehat{B}}_2(X,Y):=P_2{\nabla }_{P_2^{*}Y}{P}_1^{*}X,\hfill \\ \hfill & & {\check{B}}_1(Y,X):=P_1{\nabla }_{P_{1}X}{P}_2^{*}Y,{\check{B}}_2(X,Y):=P_2{\nabla }_{P_{2}Y}{P}_1^{*}X.\hfill \end{array}$

Note that, for self-adjoint endomorphisms $P_i$, we have $B_i={\widehat{B}}_i={\check{B}}_i$ and ${\mathcal{D}}_i^{*}={\mathcal{D}}_i$. A pair $(P_1,P_2)$ (or a tensor $P=P_1+P_2$) is allowed for a connection ∇ if $b_j^{\left(i\right)}=0,i,j\in \{1,2\}$, where the bilinear forms $b_1^{\left(i\right)}:{\left({\mathcal{X}}_M\right)}^2\to P_2\left({\mathcal{X}}_M\right)$ and their dual ones $b_2^{\left(i\right)}:{\left({\mathcal{X}}_M\right)}^2\to P_1\left({\mathcal{X}}_M\right)$ are given by

$\begin{array}{cc}\hfill & b_1^{\left(1\right)}(X,Y)=P_2^{*}{P}_2{\nabla }_{P_{1}X}{P}_1^{*}Y-P_2^{*}{\nabla }_{P_{1}X}{P}_1{P}_1^{*}Y,b_1^{\left(2\right)}(X,Y)=P_2^{*}{P}_2{\nabla }_{P_{1}X}{P}_1^{*}Y-P_2{\nabla }_{P_1^{*}{P}_{1}X}{P}_1^{*}Y,\hfill \\ \hfill & b_2^{\left(1\right)}(X,Y)=P_1^{*}{P}_1{\nabla }_{P_{2}X}{P}_2^{*}Y-P_1^{*}{\nabla }_{P_{2}X}{P}_2{P}_2^{*}Y,b_2^{\left(2\right)}(X,Y)=P_1^{*}{P}_1{\nabla }_{P_{2}X}{P}_2^{*}Y-P_1{\nabla }_{P_2^{*}{P}_{2}X}{P}_2^{*}Y.\hfill \end{array}$

An example of an allowed endomorphism is $P=f\mathrm{id}$, where $P=f{P}_1+f{P}_2$, $P_i$ are orthoprojectors onto complementary orthogonal distributions, id is the identity endomorphism of $TM$ and a function $f:M\to \mathbb{R}$ is supported on a dense set in M.

Define maps $R^P,S_i,{\mathcal{T}}_i:{\left({\mathcal{X}}_M\right)}^4\to C^{\infty }\left(M\right)$, $i\in \{1,2\}$ by

$\begin{array}{ccc}\hfill {\mathcal{T}}_1(Y,X_1,X_2,Z)& =& ⟨P_2{\nabla }_{P_1{X}_1}{B}_2(X_2,Y)-{\check{B}}_2({\nabla }_{P_1{X}_1}{P}_1{X}_2,Y)-{\widehat{B}}_2(X_2,{\nabla }_{P_1{X}_1}{P}_{2}Y),Z⟩,\hfill \\ \hfill {\mathcal{T}}_2(Y,X_1,X_2,Z)& =& ⟨P_1{\nabla }_{P_{2}Y}{B}_1(Z,X_1)-{\check{B}}_1({\nabla }_{P_{2}Y}{P}_{2}Z,X_1)-{\widehat{B}}_1(Z,{\nabla }_{P_{2}Y}{P}_1{X}_1),X_2⟩,\hfill \\ \hfill S_1(Y,X_1,X_2,Z)& =& ⟨{\widehat{B}}_2(X_2,{\nabla }_{P_{2}Y}{P}_1{X}_1),Z⟩,S_2(Y,X_1,X_2,Z)=⟨{\widehat{B}}_1(Z,{\nabla }_{P_1{X}_1}{P}_{2}Y),X_2⟩,\hfill \\ \hfill R^P(Y,X_1,X_2,Z)& =& ⟨P_2^{*}{\nabla }_{P_{2}Y}{P}_2{\nabla }_{P_1{X}_1}{P}_1^{*}{X}_2+P_2{\nabla }_{P_{2}Y}{P}_1^{*}{\nabla }_{P_1{X}_1}{P}_1{X}_2\hfill \\ & & -P_2^{*}{\nabla }_{P_1{X}_1}{P}_1{\nabla }_{P_{2}Y}{P}_1^{*}{X}_2-P_2{\nabla }_{P_1{X}_1}{P}_2^{*}{\nabla }_{P_{2}Y}{P}_1{X}_2-P_2{\nabla }_{P^{*}[P_{2}Y,P_1{X}_1]}{P}_1^{*}{X}_2,Z⟩.\hfill \end{array}$

Proposition 1

(see [21]). If $P=P_1+P_2$ is allowed for the Levi–Civita connection, then $S_i,{\mathcal{T}}_i$ and $R^P$ are tensor fields and the following Codazzi-type equation holds:

(20)$S_1+{\mathcal{T}}_1+S_2+{\mathcal{T}}_2+R^P=0.$

Remark 1.

Let $P_1$ and $P_2$ be orthoprojectors from $TM$ onto complementary orthogonal regular distributions ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$, respectively. Then, (20) reads as the Codazzi equation

(21) $\begin{array}{c}R(Y,X_1,X_2,Z)+⟨\left({\nabla }_{X_1}{B}_2\right)(X_2,Y),Z⟩+⟨\left({\nabla }_Y{B}_1\right)(Z,X_1),X_2⟩\hfill \\ +⟨B_2(X_2,B_2(X_1,Y)),Z⟩+⟨B_1(Z,B_1(Y,X_1)),X_2⟩=0,\hfill \end{array}$

where $X_i\in {\mathcal{D}}_1$ and $Y,Z\in {\mathcal{D}}_2$. For $B_1=0$ and $X_i=X$, (21) yields the Riccati equation

$\left({\nabla }_X{B}_2\right)(X,Y)+B_2(X,B_2(X,Y))+R(Y,X)X=0.$

The P-divergence of a $(1,k)$-tensor S is a $(0,k)$-tensor ${\mathrm{div}}_{P}S=\mathrm{trace}(Y\to P^{*}{\nabla }_{PY}S)$, e.g., for a vector field X on M, we obtain ${\mathrm{div}}_{P}X=\mathrm{trace}(Y\to P^{*}{\nabla }_{PY}X)$—a function on M.

Proposition 2

(see [21]). For a given $P\in \mathrm{End}\left(TM\right)$, condition

(22)$\mathrm{div}\left(P{P}^{*}\right)=0$

provides the following: ${\mathrm{div}}_{P}X=\mathrm{div}\left(P{P}^{*}\left(X\right)\right)$ for $X\in {\mathcal{X}}_M$.

Using Proposition 2, we obtain the following.

Theorem 6

(see [21]). If (22) is valid on a compact Riemannian manifold $(M,g)$, then, for any $X\in {\mathcal{X}}_M$,

${\int }_M\left({\mathrm{div}}_{P}X\right)d \mathrm{vol}={\int }_{\partial M}⟨X,P{P}^{*}\left(\nu \right)⟩d\omega .$

Consider a modified Stokes’ theorem—see also (7)—on a complete open Riemannian space.

Proposition 3.

Let a complete open Riemannian manifold $(M,g)$ be endowed with a vector field X such that ${\mathrm{div}}_{P}X\ge 0$ (or $\le 0)$, where $P\in \mathrm{End}\left(TM\right)$ satisfies (22) and $\parallel P{P}^{*}{\left(X\right)\parallel }_g\in {\mathrm{L}}^1(M,g)$; then, ${\mathrm{div}}_{P}X\equiv 0$.

If $P_i(i=1,2)$ are self-adjoint operators for an adapted Riemannian metric, then (19) means the orthogonality of distributions $P_i\left(TM\right)$, and the definition of $R^P$ reads as

$\begin{array}{c}\hfill R^P(Y,X_1,X_2,Z)=⟨P_2({\nabla }_{P_{2}Y}P{\nabla }_{P_1{X}_1}-{\nabla }_{P_1{X}_1}P{\nabla }_{P_{2}Y}-{\nabla }_{P[P_{2}Y,P_1{X}_1]})P_1{X}_2,Z⟩.\end{array}$

The second fundamental forms $h_i$ and the integrability tensors $T_i$ of singular distributions are

$\begin{array}{c}\hfill h_1(X,Y)=\frac{1}{2}{P}_2({\nabla }_{P_{1}X}{P}_{1}Y+{\nabla }_{P_{1}Y}{P}_{1}X),h_2(X,Y)=\frac{1}{2}{P}_1({\nabla }_{P_{2}X}{P}_{2}Y+{\nabla }_{P_{2}Y}{P}_{2}X),\\ \hfill T_1(X,Y)=\frac{1}{2}{P}_2\left({\nabla }_{P_{1}X}{P}_{1}Y-{\nabla }_{P_{1}Y}{P}_{1}X\right),T_2(X,Y)=\frac{1}{2}{P}_1\left({\nabla }_{P_{2}X}{P}_{2}Y-{\nabla }_{P_{2}Y}{P}_{2}X\right).\end{array}$

The mean curvature vectors of ${\mathcal{D}}_i$ are given by $H_i={\mathrm{Trace}}_g{h}_i(i=1,2)$. A distribution is called totally geodesic if its second fundamental form vanishes, and the distribution is called autoparallel if its second fundamental form and the integrability tensor simultaneously vanish; see [21] (for regular case, see [20]). A distribution is called harmonic if its mean curvature vector vanishes.

The square of the P-norm of a vector $X\in P_1\left(TM\right)\cup P_2\left(TM\right)$ is defined by

(23)${\parallel X\parallel }_P^2=\left\{\begin{array}{cc}⟨P_1\left(X^{\prime }\right),X^{\prime }⟩& \mathrm{if}X=P_1\left(X^{\prime }\right)\in P_1\left(TM\right),\\ ⟨P_2\left(X^{\prime }\right),X^{\prime }⟩& \mathrm{if}X=P_2\left(X^{\prime }\right)\in P_2\left(TM\right).\end{array}\right.$

Since the value of ${\parallel X\parallel }_P^2$ for general endomorphism $P=P_1+P_2$ is not positive, we will not use it without its square. In particular, using (23), we obtain

$\begin{array}{c}\hfill \parallel H_2{\parallel }_P^2={\sum }_{s,t}⟨P_1{\nabla }_{P_2{e}_s}{P}_2{e}_s,{\nabla }_{P_2{e}_t}{P}_2{e}_t⟩,{\parallel H_1\parallel }_P^2={\sum }_{s,t}⟨P_2{\nabla }_{P_1{e}_s}{P}_1{e}_s,{\nabla }_{P_1{e}_t}{P}_1{e}_t⟩,\end{array}$

which makes sense, since $H_1\in P_2\left(TM\right)$ and $H_2\in P_1\left(TM\right)$. Similarly, using $h_1=P_2{h}_1^{\prime }$ and $T_1=P_2{T}_1^{\prime }$, etc., we define $\parallel h_1{\parallel }_P^2={\sum }_{s,t}{\parallel h_1(e_s,e_t)\parallel }_P^2$, $\parallel T_1{\parallel }_P^2={\sum }_{s,t}{\parallel T_1(e_s,e_t)\parallel }_P^2$, etc.

Define the mixed scalar curvature of $(P_1,P_2)$ by $S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)={\sum }_{s,t}{R}^P(e_t,e_s,e_s,e_t)$, where $\left\{e_s\right\}$ is a local orthonormal frame in M, and it coincides with $S_{\mathrm{mix}}({\mathcal{D}}_1,{\mathcal{D}}_2)$ for the case of a regular almost product structure. For self-adjoint $P_1,P_2\in \mathrm{End}\left(TM\right)$, set $P=P_1+P_2$. Then,

(24)$\begin{array}{c}\hfill {\mathrm{div}}_P(H_1+H_2)=S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)+\parallel h_1{\parallel }_P^2+\parallel h_2{\parallel }_P^2-\parallel T_1{\parallel }_P^2-\parallel T_2{\parallel }_P^2-\parallel H_1{\parallel }_P^2-{\parallel H_2\parallel }_P^2.\end{array}$

The integrated P-divergence of a vector field on a closed manifold vanishes if (22) holds; see Theorem 6. For a self-adjoint P, the integral of the right hand side of (24) vanishes under a certain assumption.

Theorem 7

(see [21]). For a self-adjoint $P_i\in \mathrm{End}\left(TM\right)$$(i=1,2)$ on a closed Riemannian manifold $(M,g)$, set $P=P_1+P_2$ and assume that $\mathrm{div}\left(P^2\right)=0$.

Then, the following integral formula with ${\mathcal{D}}_i=P_i\left(TM\right)$ holds:

${\int }_M(S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)+\parallel h_1{\parallel }_P^2+\parallel h_2{\parallel }_P^2-\parallel T_1{\parallel }_P^2-\parallel T_2{\parallel }_P^2-\parallel H_1{\parallel }_P^2-{\parallel H_2\parallel }_P^2)\mathrm{d} \mathrm{vol}=0.$

Suppose that endomorphisms $P_i$ are self-adjoint and non-negative. The next results on autoparallel distributions yield splitting results in a regular case.

Theorem 8

(see [21]). Let a complete open Riemannian manifold $(M,g)$ satisfying $T_i=0$ be equipped with harmonic distributions ${\mathcal{D}}_i=P_i\left(TM\right)$. If $S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)\ge 0$, then $S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)\equiv 0$ and ${\mathcal{D}}_i$ are autoparallel.

Proof. 

By the assumptions, (24) yields ${\mathrm{div}}_P(H_1+H_2)=S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)+\parallel h_1{\parallel }_P^2+{\parallel h_2\parallel }_P^2$. Since $S_{\mathrm{mix}}^P({\mathcal{D}}_1,{\mathcal{D}}_2)\ge 0$ (and $P_i\ge 0$), by Proposition 3, we obtain ${\mathrm{div}}_P(H_1+H_2)=0$. Thus, $h_1=h_2=0$. □

3. Riemannian Almost Multi-Product Structure

In Section 3.1, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure—see [30]—and obtain splitting theorems with this curvature, which generalize (9). In Section 3.2, we represent integral formulas to hypersurfaces of space forms with $k\ge 2$ distinct principal curvatures of constant multiplicities; see [16,30]. In Section 3.3, the mixed scalar curvature is defined for an almost multi-product structure equipped with a linear (e.g., statistical) connection, and this kind of curvature is represented using the divergence of a geometrically interesting vector field. With the help of this formula, we obtain decomposition and non-existence theorems, as well as integral formulas that extend results for $k=2$ and results on Riemannian almost product manifolds; see [31]. Section 3.4 discusses integral formulas for a foliation $\mathcal{F}$ equipped with a unit normal vector field N; as a consequence, a generalization of integral formulas due to Brito–Langevin–Rosenberg and Andrzejewski–Walczak is obtained. These integral formulas deal with Newton transformations of the shape operator of $\mathcal{F}$ with respect to N and the curvature tensor of the induced connection on the distribution $\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}\left(N\right)$, and $\mathcal{F}$ can be regarded as a codimension one foliation of a sub-Riemannian manifold $(M,g,\mathcal{D})$; see [24]. In Section 3.5, we discuss the following structure on a Riemannian manifold: a distribution $\mathcal{D}$ on the tangent bundle decomposed into the sum of two orthogonal distributions, $\mathcal{D}={\mathcal{D}}_1\oplus {\mathcal{D}}_2$. This decomposition in the case of integrable ${\mathcal{D}}_1$ can be regarded as a foliated sub-Riemannian manifold. We represent integral formulas with the mixed scalar curvature of this structure—see [32,33]—that generalize results on foliations in Section 2.2.

3.1. Almost Multi-Product Structure

Let $(M,g)$ be an n-dimensional Riemannian manifold with the Levi–Civita connection ∇ and the curvature tensor R, and let ${\mathcal{D}}_i(1\le i\le k)$ be $k\ge 2$ pairwise orthogonal $n_i$-dimensional distributions of ranks $n_i$ on M.

There always exists a local adapted orthonormal frame $\{E_1,\dots ,E_n\}$ on M, where

$\{E_1,\dots ,E_{n_1}\}\subset {\mathcal{D}}_1$, $\{E_{n_1+1},\dots ,E_{n_2}\}\subset {\mathcal{D}}_2,\dots \{E_{n_{k-1}+1},\dots ,E_{n_k}\}\subset {\mathcal{D}}_k$.

Definition 1.

The function ${\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}={\sum }_{i<j}{\mathrm{S}}_{\mathrm{mix}}({\mathcal{D}}_i,{\mathcal{D}}_j)$ on $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$, where ${\mathrm{S}}_{\mathrm{mix}}({\mathcal{D}}_i,{\mathcal{D}}_j)={\sum }_{n_{i-1}<a\le n_i,n_{j-1}<b\le n_j}⟨R(E_a,E_b)E_a,E_b⟩$ and $i\ne j$, is called the mixed scalar curvature of a Riemannian almost multi-product structure $(M,g,{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$.

Let $P_i:TM\to {\mathcal{D}}_i$ be orthoprojectors and the tensors $h_i,H_i,T_j$ be related to the distribution ${\mathcal{D}}_i$. Let also $h_{ij},H_{ij},T_{ij}$ be the second fundamental forms, their mean curvature vector fields and integrability tensors associated with the distributions ${\mathcal{D}}_i\oplus {\mathcal{D}}_j$. Then, $H_i={\sum }_{j\ne i}{P}_j{H}_i$, etc. Using Definition 1, one can obtain the decomposition formula

(25)$2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}={\sum }_i{\mathrm{S}}_{{\mathcal{D}}_i,{\mathcal{D}}_i^{\perp }}.$

From (25) and (10), we obtain

$\mathrm{div}{\sum }_i\left(H_i+H_i^{\perp }\right)=2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}+{\sum }_i(\parallel h_i{\parallel }^2-\parallel H_i{\parallel }^2-\parallel T_i{\parallel }^2+\parallel h_i^{\perp }{\parallel }^2-\parallel H_i^{\perp }{\parallel }^2-{\parallel T_i^{\perp }\parallel }^2).$

Note that the scalar curvature $\mathrm{S}:M\to \mathbb{R}$ of $(M,g)$ can be represented as $\mathrm{S}=2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}+{\sum }_i\mathrm{S}\left({\mathcal{D}}_i\right)$, where $\mathrm{S}\left({\mathcal{D}}_i\right)$ are scalar curvatures of suitable distributions (functions on M). Thus, if $n_1=\dots =n_k=1$, then ${\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}$ reduces to the scalar curvature divided by two.

Denote by $S(r,k)$ the set of r-combinations (subsets of r distinct elements) of $\{1,\dots ,k\}$. The $S(r,k)$ consists of $\frac{k!}{r!(k-r)!}$ elements. For example, $S(k-1,k)$ has k elements. For $q=(q_1,\dots ,q_r)\in S(r,k)$, we can assume that $q_1<\dots <q_r$. Put $h_q=h_{q_1,\dots ,q_r}$ and $T_q=T_{q_1,\dots ,q_r}$ for ${\mathcal{D}}_{q_1}\oplus \dots \oplus {\mathcal{D}}_{q_r}$.

Theorem 9

(see [30]). For $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$, we have

(26)$\begin{array}{ccc}\hfill \mathrm{div}\xi & =& 2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}+{\sum }_{r\in \{1,k-1\}}{\sum }_{q\in S(r,k)}(\parallel h_q{\parallel }^2-\parallel H_q{\parallel }^2-{\parallel T_q\parallel }^2),\hfill \end{array}$

where $\xi ={\sum }_{r\in \{1,k-1\}}{\sum }_{q\in S(r,k)}{H}_q$, and, for a closed manifold M, we obtain the integral formula

(27)${\int }_M(2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}+{\sum }_{r\in \{1,k-1\}}{\sum }_{q\in S(r,k)}(\parallel h_q{\parallel }^2-\parallel H_q{\parallel }^2-\parallel T_q{\parallel }^2))\mathrm{d}{\mathrm{vol}}_g=0.$

If distributions ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$ are totally umbilical, then, for any $q\in S(r,k)$ with $r\ge 1$, we obtain $\parallel h_q{\parallel }^2-\parallel H_q{\parallel }^2=-{\sum }_{i=1}^r\frac{n_{q_i}-1}{n_{q_i}}{\parallel P_q^{\perp }{H}_{q_i}\parallel }^2$, where $P_q^{\perp }:TM\to {⨁}_{j\notin q}{\mathcal{D}}_j$ is the orthoprojector; in this case, (26) reduces to

$\mathrm{div}\xi =2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}-{\sum }_{r\in \{1,k-1\}}{\sum }_{q\in S(r,k)}{\sum }_{i=1}^r\left(\frac{n_{q_i}-1}{n_{q_i}}\parallel P_q^{\perp }{H}_{q_i}{\parallel }^2+{\parallel T_q\parallel }^2\right).$

For $k=3$, the integral formula (27) was proved in [34] by direct computations of $\mathrm{div}(H_1+H_2+H_3)$.

3.2. Hypersurfaces with k Distinct Principal Curvatures

Let M be a transversely orientable hypersurface in a Riemannian manifold $({\overline{M}}^{n+1},\overline{g})$, and N be a unit normal vector field to M. Let ${\lambda }_1\le \dots \le {\lambda }_n$ be the principal curvatures (eigenvalues of the shape operator $A_N$ with respect to N)—continuous functions on M. An example is a Dupin hypersurface of a real space form, defined by the following conditions: (a) along each curvature surface M, the corresponding principal curvature is constant; (b) the number of distinct principal curvatures is constant on M. A compact Dupin hypersurface M in a space form $\overline{M}\left(c\right)(c\ge 0)$ has one, two, three, four or six distinct principal curvatures, e.g., [35]; in this case, we have $(M,g,{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ with $k=2,3,4,6$ and induced metric g. For a totally umbilical hypersurface M, all principal curvatures are equal at any point of M. Let there exist $k\ge 2$ distinct principal curvatures $\left({\mu }_i\right)$ on M of multiplicities $n_i$,

${\mu }_1:={\lambda }_1=\dots ={\lambda }_{n_1}<\dots <{\mu }_k:={\lambda }_{n-n_k+1}=\dots ={\lambda }_n,$

which correspond to eigen-distributions ${\mathcal{D}}_i(1\le i\le k)$—vector sub-bundles of $TM$. In this case, we obtain a Riemannian almost multi-product manifold $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$.

For a hypersurface M with induced metric g in a Riemannian manifold ${\overline{M}}^{n+1}\left(c\right)$ of constant curvature c, the following is known; see [35]: any function ${\mu }_i:M\to \mathbb{R}$ with $n_i>1$ is differentiable and ${\mathcal{D}}_i$ defines a totally umbilical foliation of $\overline{M}\left(c\right)$. Let $X_i\in {\mathrm{X}}_i$ be local unit vector fields on ${\mathcal{D}}_i(i\le k)$. Since the curvature tensor $\overline{R}$ of $\overline{M}\left(c\right)$ has the well-known view $\overline{R}(X,Y)Z=c(⟨X,Z⟩Y-⟨Y,Z⟩X)$, for the sectional curvature of $(M,g)$, we obtain $K(X_i,X_j)=c+{\mu }_i{\mu }_j(i\ne j)$.

For a hypersurface $M^n$ in ${\overline{M}}^{n+1}\left(c\right)$ with $k=2$, using equalities

$\parallel h_i{\parallel }^2-{\parallel H_i\parallel }^2=n_i(1-n_i)\frac{\parallel \nabla {\mu }_i{\parallel }^2}{{({\mu }_i-{\mu }_j)}^2},{\mathrm{S}}_{{\mathcal{D}}_1,{\mathcal{D}}_2}=n_1{n}_2(c+{\mu }_1{\mu }_2),$

Formula (10) can be written in terms of ${\mu }_i(i=1,2)$ as follows (see [16]):

(28)$\mathrm{div}(H_1+H_2)=n_1{n}_2(c+{\mu }_1{\mu }_2)+\frac{n_1(1-n_1)\parallel \nabla {\mu }_1{\parallel }^2+n_2(1-n_2){\parallel \nabla {\mu }_2\parallel }^2}{{({\mu }_2-{\mu }_1)}^2}.$

Applying Stokes’ theorem to (28) on a closed M, we obtain the following integral formula:

$\begin{array}{c}\hfill {\int }_M\left[n_1{n}_2(c+{\mu }_1{\mu }_2)+\frac{n_1(1-n_1)\parallel \nabla {\mu }_1{\parallel }^2+n_2(1-n_2){\parallel \nabla {\mu }_2\parallel }^2}{{({\mu }_2-{\mu }_1)}^2}\right]\mathrm{d}{\mathrm{vol}}_g=0.\end{array}$

For a hypersurface M in ${\overline{M}}^{n+1}\left(c\right)$ with $k=3$ distinct principal curvatures, we have

(29)$\mathrm{div}\sum _i{n}_i\left(\frac{P_j\nabla {\mu }_i}{{\mu }_i-{\mu }_j}+\frac{P_l\nabla {\mu }_i}{{\mu }_i-{\mu }_l}\right)=\frac{1}{2}\sum _{j<l}{n}_j{n}_l(c+{\mu }_j{\mu }_l)+\sum _i{n}_i(1-n_i)\left(\frac{\parallel P_j\nabla {\mu }_i{\parallel }^2}{{({\mu }_i-{\mu }_j)}^2}+\frac{\parallel P_l\nabla {\mu }_i{\parallel }^2}{{({\mu }_i-{\mu }_l)}^2}\right),$

where $(i,j,l)\in \left\{\right(1,2,3),(2,1,3),(3,1,2\left)\right\}$. Applying Stokes’ theorem to (29) on a closed M, we obtain the following integral formula (see [30]):

${\int }_M[\sum _{j<l}{n}_j{n}_l(c+{\mu }_j{\mu }_l)+2\sum _i{n}_i(1-n_i)(\frac{\parallel P_j\nabla {\mu }_i{\parallel }^2}{{({\mu }_i-{\mu }_j)}^2}+\frac{\parallel P_l\nabla {\mu }_i{\parallel }^2}{{({\mu }_i-{\mu }_l)}^2}\left)\right]\mathrm{d}{\mathrm{vol}}_g=0.$

One can try to find formulas analogous to (28) and (29) for a hypersurface in ${\overline{M}}^{n+1}\left(c\right)$ with $k\in \{4,6\}$ distinct principal curvatures of constant multiplicities.

3.3. Metric-Affine Manifolds

For a Riemannian manifold $(M^m,g)$ with a linear connection $\overline{\nabla }$ and the curvature tensor $\overline{R}$, equipped with two orthogonal complementary distributions $(\mathcal{D},{\mathcal{D}}^{\perp })$, the mixed scalar curvature is defined by

${\overline{\mathrm{S}}}_{\mathcal{D},{\mathcal{D}}^{\perp }}=\frac{1}{2}\sum _{1\le a\le n,n<b\le m}\left(⟨{\overline{R}}_{e_a,e_b}{e}_a,e_b⟩+⟨{\overline{R}}_{e_b,e_a}{e}_b,e_a⟩\right).$

Here, $\{e_1,\dots ,e_m\}$ is a local adapted orthonormal frame on $(M,g)$, i.e., $e_a\in \mathcal{D}$ for $1\le a\le n=\dim \mathcal{D}$. For $\mathcal{D}$ spanned by a unit vector field N, we obtain ${\overline{\mathrm{S}}}_{\mathcal{D},{\mathcal{D}}^{\perp }}={\overline{\mathrm{Ric}}}_{N,N}$.

The following function on M is called the mixed scalar curvature with respect to $\overline{\nabla }$ on $(M,g,\overline{\nabla };{\mathcal{D}}_1,\dots {\mathcal{D}}_k)$:

${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}=\frac{1}{2}\sum _{i<j}\sum _{n_{i-1}<a\le n_i,n_{j-1}<b\le n_j}\left(⟨{\overline{R}}_{e_a,e_b}{e}_a,e_b⟩+⟨{\overline{R}}_{e_b,e_a}{e}_b,e_a⟩\right).$

Here, $\{e_1,\dots ,e_m\}$ is again a local adapted orthonormal frame on M, but now $\{e_1,\dots ,e_{n_1}\}\subset {\mathcal{D}}_1$ and $\{e_{n_{i-1}+1},\dots ,e_{n_i}\}\subset {\mathcal{D}}_i$ for $i\ge 2$. In particular, if $\mathrm{T}=0$, then ${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}$ reduces to the mixed scalar curvature of $(M,g;{\mathcal{D}}_1,\dots {\mathcal{D}}_k)$ with respect to the Levi–Civita connection ∇,

${\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}=\sum _{i<j}\sum _{n_{i-1}<a\le n_i,n_{j-1}<b\le n_j}⟨R_{e_a,e_b}{e}_a,e_b⟩,$

defined as an averaged mixed sectional curvature. Observe that the scalar curvature $\overline{\mathrm{S}}$ is decomposed as $\overline{\mathrm{S}}=2{\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}+{\sum }_i\overline{\mathrm{S}}{|}_{{\mathcal{D}}_i}$, where $\overline{\mathrm{S}}{|}_{{\mathcal{D}}_i}$ is the scalar curvature of $(M,g)$ along the plane field ${\mathcal{D}}_i$. For any $(M,g,\overline{\nabla };{\mathcal{D}}_1,\dots {\mathcal{D}}_k)$, we have the following decomposition of ${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}$ similar to (25):

$2{\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}={\sum }_i{\overline{\mathrm{S}}}_{{\mathcal{D}}_i,{\mathcal{D}}_i^{\perp }}.$

Define the partial traces of a contorsion tensor $\mathrm{T}$ by

${\mathrm{trace}}_{{\mathcal{D}}_i^{\perp }}\mathrm{T}={\sum }_{b\ne (n_{i-1},n_i]}{\mathrm{T}}_{e_b}{e}_b,{\mathrm{trace}}_{{\mathcal{D}}_i}\mathrm{T}={\sum }_{n_{i-1}<a\le n_i}{\mathrm{T}}_{e_a}{e}_a.$

For $(M,g,\overline{\nabla }=\nabla +\mathrm{T};\mathcal{D},{\mathcal{D}}^{\perp })$, one can obtain

(30)$\begin{array}{c}\hfill \begin{array}{ccc}& & \hfill \mathrm{div}\left(P{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}(\mathrm{T}-{\mathrm{T}}^{*})+P^{\perp }{\mathrm{trace}}_{\mathcal{D}}(\mathrm{T}-{\mathrm{T}}^{*})\right)=2({\overline{\mathrm{S}}}_{\mathcal{D},{\mathcal{D}}^{\perp }}-{\mathrm{S}}_{\mathcal{D},{\mathcal{D}}^{\perp }})-⟨{\mathrm{trace}}_{\mathcal{D}}\mathrm{T},{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}{\mathrm{T}}^{*}⟩\\ & & \hfill -⟨{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}\mathrm{T},{\mathrm{trace}}_{\mathcal{D}}{\mathrm{T}}^{*}⟩-⟨{\mathrm{trace}}_{\mathcal{D}}(\mathrm{T}-{\mathrm{T}}^{*})-{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}(\mathrm{T}-{\mathrm{T}}^{*}),H-H^{\perp }⟩\\ & & \hfill -⟨\mathrm{T}-{\mathrm{T}}^{*}+{\mathrm{T}}^{\wedge }-{\mathrm{T}}^{*\wedge },A^{\perp }-T^{\perp ♯}+A-T^{♯}⟩+{⟨{\mathrm{T}}^{*},{\mathrm{T}}^{\wedge }⟩}_{\left|V\right(\mathcal{D})}.\end{array}\end{array}$

The auxiliary functions $Q(\mathcal{D},g)$ and $\overline{Q}(\mathcal{D},g,\mathrm{T})$ are used and given by

(31)$\begin{array}{ccc}\hfill Q(\mathcal{D},g)& =& ⟨H^{\perp },H^{\perp }⟩+⟨H,H⟩-⟨h,h⟩-⟨h^{\perp },h^{\perp }⟩+⟨T,T⟩+⟨T^{\perp },T^{\perp }⟩,\hfill \\ \hfill 2\overline{Q}(\mathcal{D},g,\mathrm{T})& =& ⟨{\mathrm{trace}}_{\mathcal{D}}\mathrm{T},{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}{\mathrm{T}}^{*}⟩+⟨{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}\mathrm{T},{\mathrm{trace}}_{\mathcal{D}}{\mathrm{T}}^{*}⟩+⟨{\mathrm{trace}}_{\mathcal{D}}(\mathrm{T}-{\mathrm{T}}^{*})\hfill \end{array}$

(32)$\begin{array}{ccc}& & -{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}(\mathrm{T}-{\mathrm{T}}^{*}),H-H^{\perp }⟩-{⟨{\mathrm{T}}^{*},{\mathrm{T}}^{\wedge }⟩}_{\left|V\right(\mathcal{D})}+⟨\mathrm{T}-{\mathrm{T}}^{*}+{\mathrm{T}}^{\wedge }-{\mathrm{T}}^{*\wedge },A^{\perp }-T^{\perp ♯}+A-T^{♯}⟩.\hfill \end{array}$

Formulas (10) and (30) can be written shortly as

(33)$\begin{array}{ccc}\hfill \mathrm{div}(H+H^{\perp })& =& {\mathrm{S}}_{\mathcal{D},{\mathcal{D}}^{\perp }}-Q(\mathcal{D},g),\hfill \end{array}$

(34)$\begin{array}{ccc}\hfill \mathrm{div}\left(P{\mathrm{trace}}_{{\mathcal{D}}^{\perp }}(\mathrm{T}-{\mathrm{T}}^{*})+P^{\perp }{\mathrm{trace}}_{\mathcal{D}}(\mathrm{T}-{\mathrm{T}}^{*})\right)& =& 2({\overline{\mathrm{S}}}_{\mathcal{D},{\mathcal{D}}^{\perp }}-{\mathrm{S}}_{\mathcal{D},{\mathcal{D}}^{\perp }})-2\overline{Q}(\mathcal{D},g,\mathrm{T}).\hfill \end{array}$

The last terms in (32) and ${⟨{\mathrm{T}}^{*},{\mathrm{T}}^{\wedge }⟩}_{\left|V\right(\mathcal{D})}$ have the following form in a local adapted basis:

$\begin{array}{c}⟨\mathrm{T}-{\mathrm{T}}^{*}+{\mathrm{T}}^{\wedge }-{\mathrm{T}}^{*\wedge },A^{\perp }-T^{\perp ♯}+A-T^{♯}⟩\hfill \\ ={\sum }_{a\le n_1,b>n_1}(⟨({\mathrm{T}}_{e_b}-{\mathrm{T}}_{e_b}^{*})e_a+({\mathrm{T}}_{e_a}-{\mathrm{T}}_{e_a}^{*})e_b,(A_{e_a}^{\perp }-T_{e_a}^{\perp ♯})e_b+(A_{e_b}-T_{e_b}^{♯})e_a⟩),\hfill \\ {⟨{\mathrm{T}}^{*},{\mathrm{T}}^{\wedge }⟩}_{\left|V\right(\mathcal{D})}={\sum }_{a\le n_1,b>n_1}\left(⟨{\mathrm{T}}_{e_a}{e}_b,{\mathrm{T}}_{e_b}^{*}{e}_a⟩+⟨{\mathrm{T}}_{e_a}^{*}{e}_b,{\mathrm{T}}_{e_b}{e}_a⟩\right).\hfill \end{array}$

The following result for a linear connection $\overline{\nabla }$ instead of ∇ and $k\ge 2$ is based on (33) and (34).

Proposition 4.

Let $(M,g,\overline{\nabla };{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ be an almost multi-product manifold with a linear connection $\overline{\nabla }=\nabla +\mathrm{T}$; then,

$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{div}\xi =2{\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}-{\sum }_i\left(\overline{Q}({\mathcal{D}}_i,g,\mathrm{T})+Q({\mathcal{D}}_i,g)\right),\end{array}\end{array}$

where $\xi ={\sum }_i(\frac{1}{2}{P}_i{\mathrm{trace}}_{{\mathcal{D}}_i^{\perp }}(\mathrm{T}-{\mathrm{T}}^{*})+\frac{1}{2}{P}_i^{\perp }{\mathrm{trace}}_{{\mathcal{D}}_i}(\mathrm{T}-{\mathrm{T}}^{*})+H_i+H_i^{\perp })$, $Q({\mathcal{D}}_i,g)$ and $\overline{Q}({\mathcal{D}}_i,g,\mathrm{T})$ are given in (31) and (32) with $\mathcal{D}={\mathcal{D}}_i$. If M is a closed manifold, then the following integral formula is valid:

(35) ${\int }_M\left(2{\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}-{\sum }_i(Q({\mathcal{D}}_i,g)+\overline{Q}({\mathcal{D}}_i,g,\mathrm{T}))\right)\mathrm{d}{\mathrm{vol}}_g=0.$

For $\mathrm{T}=0$, (35) reduces to the the integral formula (27), represented equivalently as

${\int }_M\left(2{\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}-{\sum }_{i}Q({\mathcal{D}}_i,g)\right)\mathrm{d}{\mathrm{vol}}_g=0.$

For $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ with a statistical connection $\overline{\nabla }=\nabla +\mathrm{T}$, we have, for each i,

$2\overline{Q}({\mathcal{D}}_i,g,\mathrm{T})=2⟨{\mathrm{trace}}_{{\mathcal{D}}_i}\mathrm{T},{\mathrm{trace}}_{{\mathcal{D}}_i^{\perp }}\mathrm{T}⟩-{⟨\mathrm{T},\mathrm{T}⟩}_{\left|V\right({\mathcal{D}}_i)}.$

A pair of distributions, $({\mathcal{D}}_i,{\mathcal{D}}_j)$ with $i\ne j$ on $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ (with $k\ge 2$) is said to be

• (a). Mixed totally geodesic if $h_{ij}(X,Y)=0$ for $X\in {\mathcal{D}}_i$ and $Y\in {\mathcal{D}}_j$.

• (b). Mixed integrable (each is not necessarily integrable), if $T_{ij}(X,Y)=0$ for $X\in {\mathcal{D}}_i$ and $Y\in {\mathcal{D}}_j$.

A linear connection $\overline{\nabla }=\nabla +\mathrm{T}$ on $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ is called adapted if $\mathrm{T}$ is decomposed into ${\mathcal{D}}_i$-components, i.e.,

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

For an adapted statistical connection on $(M,g;{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$, one can obtain ${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}={\mathrm{S}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}$. The above definition allows us to simplify the presentation of the following results.

Theorem 10.

Let $(M,g,\overline{\nabla };{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ be an almost multi-product manifold with a statistical adapted connection $\overline{\nabla }=\nabla +\mathrm{T}$, the distributions ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$ define harmonic foliations and each pair $({\mathcal{D}}_i,{\mathcal{D}}_j)$ be a mixed integrable. If the inequality ${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}\ge 0$ holds, then $(M,g)$ splits.

For a totally umbilical distribution ${\mathcal{D}}_i$, we obtain $\parallel h_i{\parallel }^2-\parallel H_i{\parallel }^2=-\frac{n_i-1}{n_i}{\parallel H_i\parallel }^2$, and, thus, the following.

Theorem 11.

Let $(M,g,\overline{\nabla };{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k)$ be an almost multi-product manifold with a statistical adapted connection $\overline{\nabla }=\nabla +\mathrm{T}$, the distributions ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$ be totally umbilical, each pair $({\mathcal{D}}_i,{\mathcal{D}}_j)$ be mixed totally geodesic and $⟨H_i,H_j⟩=0$ for all $i\ne j$. Suppose that $(M,g)$ is completely open, $\parallel \xi \parallel \in {\mathrm{L}}^1(M,g)$ for $\xi ={\sum }_i(H_i+H_i^{\perp })$ and ${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}\le 0$; then, $(M,g)$ splits.

Corollary 2.

Let a multiply twisted product manifold of $k>2$ Riemannian manifolds be compact and equipped with a statistical adapted connection and let $⟨H_i,H_j⟩=0$ for $i\ne j$. If ${\overline{\mathrm{S}}}_{{\mathcal{D}}_1,\dots ,{\mathcal{D}}_k}\le 0$, then the manifold is the metric product (since it is diffeomorphic to the product).

3.4. Foliations Equipped with a Unit Normal Vector Field

One can ask: can we find foliations of arbitrary codimension integral formulas similar to (2)–(6)? We studied this question for a foliation $\mathcal{F}$, equipped with a unit normal vector field N. Let a distribution $\mathcal{D}$ on M be spanned by $T\mathcal{F}$ (the tangent bundle of $\mathcal{F}$) and N:

$\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}\left(N\right).$

A smooth manifold M equipped with a non-integrable distribution $\mathcal{D}$ and a Riemannian metric on $\mathcal{D}$ belongs to sub-Riemannian geometry—see [20]—and a pair $(M,\mathcal{D})$ is called a non-holonomic manifold; see [20]. Non-holonomic manifolds are used for a geometric interpretation of constrained systems in mechanics. A non-holonomic manifold $(M,\mathcal{D})$, equipped with a sub-Riemannian metric $g=⟨·,·⟩$, i.e., the scalar product $g:{\mathcal{D}}_x\times {\mathcal{D}}_x\to \mathbb{R}$ for all $x\in M$, is called a sub-Riemannian manifold. The sub-Riemannian metric g on $\mathcal{D}$ can be extended to a Riemannian metric, also denoted by g, on M. Indeed, let ${\mathcal{D}}^{\prime }$ be the normal distribution for $\mathcal{D}$ with respect to any Riemannian metric $g^{\prime }$ (which exists) on M; then, $g\oplus g^{\prime }{|}_{{\mathcal{D}}^{\prime }}$ is an extension of g onto $TM$. With this metric, we define the orthogonal distribution ${\mathcal{D}}^{\perp }$ (the vertical sub-bundle) such that $TM=\mathcal{D}\oplus {\mathcal{D}}^{\perp }$. A sub-Riemannian manifold $(M,\mathcal{D})$ equipped with a foliation such that its tangent bundle is a sub-bundle of $\mathcal{D}$ will be called a foliated sub-Riemannian manifold. Thus, we will discuss integral formulas for codimension one foliations of a sub-Riemannian manifold.

The orthoprojector $P:TM\to \mathcal{D}$ onto the regular distribution $\mathcal{D}$ satisfies the following equalities:

$P=P^{*}\left(P\mathrm{is}\mathrm{self}\text{-}\mathrm{adjoint}\right),P^2=P,$

where $P^{*}$ is the conjugate operator, e.g., [1]. The Levi–Civita connection ∇ on $(M,g)$ induces a linear connection ${\nabla }^P$ on $\mathcal{D}=P\left(TM\right)$:

${\nabla }_X^{P}PY=P{\nabla }_{X}PY,X,Y\in \Gamma \left(TM\right),$

which is metric compatible: $X⟨U,V⟩=⟨{\nabla }_X^{P}U,V⟩+⟨U,{\nabla }_X^{P}V⟩$ for any $U,V\in \Gamma \left(\mathcal{D}\right)$ and $X\in \Gamma \left(TM\right)$. The shape operator $A_N:T\mathcal{F}\to T\mathcal{F}$ of the foliation $\mathcal{F}$ with respect to N is given by

$A_N\left(X\right)=-{\nabla }_X^{P}N,X\in T\mathcal{F}.$

The generalized mean curvatures ${\sigma }_r={\sigma }_r\left(A_N\right)$ are defined similarly to (1). Our integral formulas deal with r-th mean curvatures of $\mathcal{F}$ (symmetric functions of $A_N$), the Newton transformations of $A_N$ and the curvature tensor of the induced linear connection on $\mathcal{D}$. The following theorem generalizes (2).

Theorem 12

(see [24]). Let $(M,\mathcal{D},g)$ with $\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}\left(N\right)$ be a closed sub-Riemannian manifold and the orthogonal distribution ${\mathcal{D}}^{\perp }$ be harmonic. Then, the integral formula ${\int }_M{\sigma }_1\mathrm{d}{\mathrm{vol}}_g=0$ holds; thus, ${\sigma }_1\equiv 0$ or ${\sigma }_1\left(x\right){\sigma }_1\left(x^{\prime }\right)<0$ for some points $x\ne x^{\prime }$ on M.

Using ${\nabla }^P$, define the curvature tensor $R^P:TM\times TM\to \mathrm{End}\left(\mathcal{D}\right)$ by $R_{X,Y}^P={\nabla }_X^P{\nabla }_Y^P-{\nabla }_Y^P{\nabla }_X^P-{\nabla }_{[X,Y]}^P$ and set $R^P(X,Y,V,U)=⟨R^P(X,Y)V,U⟩$ for $U,V\in \mathcal{D}$. We call $R^P(X,Y,Y,X)/(X^2{Y}^2-{⟨X,Y⟩}^2)$ a sectional P-curvature of a plane $X\wedge Y$ with non-collinear vectors. We have $R^P(Y,X)=-R^P(X,Y)$ as for any linear connection. Since ${\nabla }^P$ is metric-compatible, the anti-symmetry $R^P(X,Y,V,U)=-R^P(X,Y,U,V)$ for the last pair of vectors is valid. The following Codazzi type equation is valid for $\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}\left(N\right)$ on $(M,g)$:

$\left({\nabla }_X^{\mathcal{F}}{A}_N\right)Y-\left({\nabla }_Y^{\mathcal{F}}{A}_N\right)X=-R^P(X,Y)N,X,Y\in T\mathcal{F}.$

Define the $\mathcal{F}$-divergence ${\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_N\right)={\sum }_{i=1}^n\left({\nabla }_{e_i}^{\mathcal{F}}{T}_r\left(A_N\right)\right)e_i$ of $T_r\left(A_N\right)$—see [9]—and note that ${\mathrm{div}}_{\mathcal{F}}{T}_0\left(A_N\right)=0$. Let a linear operator ${\mathcal{R}}_X^P:T\mathcal{F}\to T\mathcal{F}(X\in \mathcal{D})$ be given by

${\mathcal{R}}_X^P:V\to R^P(V,X)N,V\in T\mathcal{F}.$

For the leafwise divergence of $T_r\left(A_N\right)$ for $r>0$, we obtain the inductive formula

$⟨{\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_N\right),X⟩=-⟨{\mathrm{div}}_{\mathcal{F}}{T}_{r-1}\left(A_N\right),A_{N}X⟩+{\mathrm{trace}}_{\mathcal{F}}(T_{r-1}\left(A_N\right){\mathcal{R}}_X^P),$

where $X\in \Gamma \left(\mathcal{F}\right)$. For $r>0$, this is equivalent to

(36)$⟨{\mathrm{div}}_{\mathcal{F}}{T}_r\left(A_N\right),X⟩={\sum }_{j=1}^r{(-1)}^{j-1}{\mathrm{trace}}_{\mathcal{F}}(T_{r-j}\left(A_N\right){\mathcal{R}}_{A_N^{j-1}X}^P).$

The following result generalizes (6).

Theorem 13

(see [24]). Let $(M,g)$ be a closed sub-Riemannian manifold with $\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}\left(N\right)$ and a harmonic orthogonal distribution ${\mathcal{D}}^{\perp }$. Then, for $r\in [0,n-2]$, we obtain

(37)${\int }_M((r+2){\sigma }_{r+2}-{\mathrm{trace}}_{\mathcal{F}}(T_r(A_N){\mathcal{R}}_N^P)-\underline{⟨{\mathrm{div}}_{\mathcal{F}}{T}_r(A_N),{\nabla }_N^{P}N⟩})\mathrm{d}{\mathrm{vol}}_g=0,$

where the underlined term is given by (36) with $X={\nabla }_N^{P}N$.

If ${\nabla }_N^{P}N=0$, i.e., N is a P-geodesic vector field, then (37) reduces to the formula ${\int }_M((r+2){\sigma }_{r+2}-{\mathrm{trace}}_{\mathcal{F}}(T_r(A_N){\mathcal{R}}_N^P))\mathrm{d}{\mathrm{vol}}_g=0$. For $r=0$, (37) simplifies the following generalization of (3):

(38)${\int }_M(2{\sigma }_2-{\mathrm{Ric}}_{N,N}^P)\mathrm{d}{\mathrm{vol}}_g=0.$

Using (38), we obtain non-existent results for P-harmonic foliations (${\sigma }_1=0$) and for P-totally umbilical foliations ($A_N=({\sigma }_1/n){\mathrm{id}}_{T\mathcal{F}}$). For a closed manifold $(M,\mathcal{D},g)$ with a harmonic orthogonal distribution ${\mathcal{D}}^{\perp }$, the following holds. (i) If ${\mathrm{Ric}}^P>0$, then P-harmonic codimension-one foliations in $\mathcal{D}$ do not exist. (ii) If ${\mathrm{Ric}}^P<0$, then P-totally umbilical codimension one foliations in $\mathcal{D}$ do not exist.

The following corollary of Theorem 13 generalizes result in ([9], Section 4.2) on codimension one totally umbilical foliations of Einstein manifolds.

Corollary 3.

Let $(M,\mathcal{D},g)$ be a closed sub-Riemannian manifold with $\mathcal{D}=T\mathcal{F}\oplus \mathrm{span}\left(N\right)$, where $\mathcal{F}$ is a P-totally umbilical foliation with $\dim \mathcal{F}>1$, a harmonic orthogonal distribution ${\mathcal{D}}^{\perp }$, and let $\mathcal{D}$ have constant P-Ricci curvature, i.e., ${\mathrm{Ric}}_{X,Y}^P=C⟨X,Y⟩$ for $X,Y\in \mathcal{D},\left|X\right|=\left|Y\right|=1$.

Then, we obtain

${\sigma }_r(\mathcal{F},N)=\{\begin{array}{cc}{(C/n)}^{r/2}\left(\begin{array}{c}n/2\\ r/2\end{array}\right)\mathrm{Vol}(M,g),& n,r\mathrm{even},\\ 0,& r\mathrm{odd}.\end{array}$

The above integral formulas can be formulated for non-integrable distributions and foliations defined outside of a “singularity set” under the assumption of the convergence of certain integrals. The results of this section were extended in [36] for a distribution $\mathcal{D}=T\mathcal{F}\oplus {\mathcal{D}}^{\prime }$ with $\dim {\mathcal{D}}^{\prime }>1$.

3.5. Distributions on Sub-Riemannian Manifolds

Consider the following structure (see [32,37]): a Riemannian manifold equipped with a distribution represented as the sum of $k>2$ pairwise orthogonal distributions. The mixed scalar curvature of this structure is involved in integral formulas generalizing results on foliations and distributions generating the tangent bundle of a manifold.

A Riemannian almost-product manifold is a Riemannian manifold $(M,g)$ equipped with a self-adjoint (1,1)-tensor P such that $P^2=P$, e.g., [1]. The orthoprojectors onto complementary orthogonal distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$ are the tensors P and ${\mathrm{id}}_{TM}-P$.

Given orthoprojector $P:TM\to \mathcal{D}$ on the regular distribution $\mathcal{D}$ on $(M,g)$, define the P-divergence of a vector field $X\in {\mathcal{X}}_M$ for any local orthonormal frame $\left\{e_s\right\}$ in M by

${\mathrm{div}}_{P}X=\mathrm{trace}(Y\to P{\nabla }_{PY}X)={\sum }_s⟨P{\nabla }_{P{e}_s}X,e_s⟩.$

Proposition 5

(see [32]). Let $(M,g)$ be a Riemannian manifold. The distributions $\mathcal{D}=P\left(TM\right)$, where $P:TM\to \mathcal{D}$ is the orthoprojector, and its orthogonal complement is harmonic if and only if $\mathrm{div}P=0$. In this case, if M is compact and without a boundary, then ${\int }_M\left({\mathrm{div}}_{P}X\right)d{\mathrm{vol}}_g=0$ for any vector field X.

Now, let $\mathcal{D}$ be the sum $\mathcal{D}={\mathcal{D}}_1\oplus {\mathcal{D}}_2$ of two orthogonal distributions on $(M,g)$, and let $P_1:TM\to {\mathcal{D}}_1$ and $P_2:TM\to {\mathcal{D}}_2$ be the orthoprojectors; hence, $P=P_1+P_2$. The following tensors are called the P-second fundamental form and the P-integrability tensor, ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$:

(39)$\begin{array}{l}{\mathcal{B}}_1(X,Y)=P_2({\nabla }_{P_{1}X}{P}_{1}Y+{\nabla }_{P_{1}Y}{P}_{1}X)/2,{\mathcal{B}}_2(X,Y)=P_1({\nabla }_{P_{2}X}{P}_{2}Y+{\nabla }_{P_{2}Y}{P}_{2}X)/2,\\ {\mathcal{T}}_1(X,Y)=P_2\left({\nabla }_{P_{1}X}{P}_{1}Y-{\nabla }_{P_{1}Y}{P}_{1}X\right)/2,{\mathcal{T}}_2(X,Y)=P_1\left({\nabla }_{P_{2}X}{P}_{2}Y-{\nabla }_{P_{2}Y}{P}_{2}X\right)/2.\end{array}$

Define the P-mean curvature vectors ${\mathcal{H}}_i={\mathrm{Trace}}_g{\mathcal{B}}_i$ of ${\mathcal{D}}_i(i=1,2)$ by

(40)${\mathcal{H}}_1={\sum }_s{P}_2\left({\nabla }_{P_1{e}_s}{P}_1{e}_s\right),{\mathcal{H}}_2={\sum }_s{P}_1\left({\nabla }_{P_2{e}_s}{P}_2{e}_s\right).$

The above definition of ${\mathcal{H}}_i$ is correct because of the orthogonality of ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$. The norms of the above tensors are given as usual by $\parallel {\mathcal{B}}_1{\parallel }^2={\sum }_{s,t}{\parallel {\mathcal{B}}_1(e_s,e_t)\parallel }^2$, $\parallel {\mathcal{T}}_1{\parallel }^2={\sum }_{s,t}{\parallel {\mathcal{T}}_1(e_s,e_t)\parallel }^2$, etc. A distribution is said to be P-totally geodesic if its P-second fundamental form vanishes, and the distribution is said to be P-integrable if its P-integrability tensor vanishes; see (39). A distribution is said to be P-harmonic if its P-mean curvature vector vanishes. We have the following relations of the second fundamental forms $h_i$, the integrability tensors $T_i$ and the mean curvature vectors $H_i={\mathrm{Trace}}_g{h}_i$ of ${\mathcal{D}}_i$ (of ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ as distributions in $TM$) with the above tensors: ${\mathcal{B}}_i=P\circ h_i$, ${\mathcal{T}}_i=P\circ T_i$ and ${\mathcal{H}}_i=P\left(H_i\right)$. Note that $\parallel {\mathcal{B}}_i\parallel \le \parallel h_i\parallel$, $\parallel {\mathcal{T}}_i\parallel \le \parallel T_i\parallel$ and $\parallel {\mathcal{H}}_i\parallel \le \parallel H_i\parallel$. The tensors ${\mathcal{B}}_i,{\mathcal{T}}_i$ and ${\mathcal{H}}_i$—see (39) and (40)—are implemented in the following results generalizing (10).

The mappings $R_i^P(X,V)W=P_i^{\perp }({\nabla }_{P_i^{\perp }X}P{\nabla }_{P_{i}V}-{\nabla }_{P_{i}V}P{\nabla }_{P_i^{\perp }X}-{\nabla }_{P[P_i^{\perp }X,P_{i}V]})P_{i}W$ are tensor fields on $(M,g)$, see [32]. The function ${\mathrm{S}}_{{\mathcal{D}}_1,{\mathcal{D}}_2}^P={\sum }_{s<t}\left(R_1^P(e_s,e_t,e_t,e_s)+R_2^P(e_s,e_t,e_t,e_s)\right)$ on M, where $\left\{e_s\right\}$ is a local orthonormal frame in $(M,g)$, will be called the mixed scalar P-curvature.

Proposition 6

(see [32]). For orthogonal distributions ${\mathcal{D}}_1=P_1\left(TM\right)$ and ${\mathcal{D}}_2=P_2\left(TM\right)$ and the orthoprojector $P=P_1+P_2$, the following formula with the mixed scalar P-curvature is satisfied:

(41)${\mathrm{div}}_P({\mathcal{H}}_1+{\mathcal{H}}_2)={\mathrm{S}}_{{\mathcal{D}}_1,{\mathcal{D}}_2}^P+\parallel {\mathcal{B}}_1{\parallel }^2+\parallel {\mathcal{B}}_2{\parallel }^2-\parallel {\mathcal{H}}_1{\parallel }^2-\parallel {\mathcal{H}}_2{\parallel }^2-\parallel {\mathcal{T}}_1{\parallel }^2-{\parallel {\mathcal{T}}_2\parallel }^2.$

Remark 2.

Define endomorphisms ${\mathcal{A}}_i$ of distributions ${\mathcal{D}}_i$ on $(M,g)$ by $⟨{\mathcal{A}}_{i}X,Y⟩=⟨{\mathcal{B}}_i(X,Y),$${\mathcal{H}}_i⟩$, $X,Y\in {\mathcal{D}}_i$. One can ask (see [14] for two complementary distributions): can we express the P-divergence of vector fields ${\left({\mathcal{A}}_1\right)}^k{\mathcal{H}}_2+{\left({\mathcal{A}}_2\right)}^k{\mathcal{H}}_1(k>0)$ in terms of the extrinsic geometry of orthogonal distributions ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$? For $k=0$, we have ${\mathcal{H}}_1+{\mathcal{H}}_2$—its P-divergence is given in (41).

Theorem 14

(see [32]). Let $(M,g)$ be a closed Riemannian manifold equipped with orthogonal distributions ${\mathcal{D}}_1=P_1\left(TM\right)$ and ${\mathcal{D}}_2=P_2\left(TM\right)$, defined on the complement to the singularity set ${\Sigma }_2$. Put $P=P_1+P_2$ and suppose that $\mathrm{div}P=0$ and $\parallel {\mathcal{H}}_1+{\mathcal{H}}_2\parallel \in L^2(M,g)$. Then,

${\int }_M\left({\mathrm{S}}_{{\mathcal{D}}_1,{\mathcal{D}}_2}^P+\parallel {\mathcal{B}}_1{\parallel }^2+\parallel {\mathcal{B}}_2{\parallel }^2-\parallel {\mathcal{H}}_1{\parallel }^2-\parallel {\mathcal{H}}_2{\parallel }^2-\parallel {\mathcal{T}}_1{\parallel }^2-{\parallel {\mathcal{T}}_2\parallel }^2\right)\mathrm{d}{\mathrm{vol}}_g=0.$

Theorem 15

(see [32]). Let $(M,g)$ be a Riemannian manifold equipped with orthogonal distributions ${\mathcal{D}}_1=P_1\left(TM\right)$ and ${\mathcal{D}}_2=P_2\left(TM\right)$. Set $P=P_1+P_2$ and let ${\mathcal{D}}_1$ be P-harmonic, ${\mathcal{D}}_2$ be P-integrable and ${\mathrm{S}}_{{\mathcal{D}}_1,{\mathcal{D}}_2}^P>0$. Then, there are no compact submanifolds of M tangent to ${\mathcal{D}}_1$.

4. Conclusions

The article reviews classical and modern results with integral formulas for almost multi-product manifolds, foliations of any codimension and multiply twisted products of Riemannian, sub-Riemannian and metric-affine manifolds. Such formulas are one of the main topic of the extrinsic geometry of foliations and are useful for studying questions such as (i) the existence and characterization of foliations, whose leaves are totally geodesic, totally umbilical or minimal; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) critical points of functionals defined for tensors on foliated Riemannian manifolds.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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