In this paper, we extend our previous study regarding a Riemannian manifold endowed with a singular (or regular) distribution, generalizing Bochner’s technique and a statistical structure. Following the construction of an almost Lie algebroid, we define the central concept of the paper: The Weitzenböck type curvature operator on tensors, prove the Bochner–Weitzenböck type formula and obtain some vanishing results about the null space of the Hodge type Laplacian on a distribution.
Distributions (subbundles of the tangent bundle) on a manifold are used to build up notions of integrability, and specifically, of a foliation, e.g., [1,2,3]. There is definite interest of pure and applied mathematicians to singular distributions and foliations, i.e., having varying dimension, e.g., [4,5]. Another popular mathematical concept is a statistical structure, i.e., a Riemannian manifold endowed with a torsionless linear connection such that the tensor is symmetric in all its entries, e.g., [6,7,8,9,10,11,12]. The theory of affine hypersurfaces in is a natural source of such manifolds; they also find applications in theory of probability and statistics as well as in information geometry.
Recall (e.g., ) that a singular distribution on a manifold M assigns to each point a linear subspace of the tangent space in such a way that, for any , there exists a smooth vector field V defined in a neighborhood U of x and such that and for all y of U. A priori, the dimension of is not constant and depends on . If , then is regular.Singular foliations are defined as families of maximal integral submanifolds (leaves) of integrable singular distributions (certainly, regular foliations correspond to integrable regular distributions). Singular distributions also arise when considering irregular mappings of manifolds, since at the point where the rank of the mapping is less than the dimension of the manifold—the inverse image, the kernel of the mapping arises. Its dimension can vary from point to point. Therefore, the theory presented in the article has applications to differential topology and mathematical analysis.
Let M be a connected smooth n-dimensional manifold, —the tangent bundle, —the Lie algebra of smooth vector fields on M, and —the space of all smooth endomorphisms of . Let be a Riemannian metric on M and ∇—the Levi–Civita connection of g.
In this paper, we apply the almost Lie algebroid structure (see a short survey in Section 8) to singular distributions on M, and in the rest of paper assume and .
(see ).An image of under a smooth endomorphism will be called a generalized vector subbundle of or a singular distribution.
(a) Let on be of constant rank, , satisfying
where is adjoint endomorphism to P, i.e., , then we have an almost product structure on , see . In this case, P and are orthoprojectors onto vertical distribution and horizontal distribution , which are complementary orthogonal and regular, but none of which is in general integrable. Many popular geometrical structures belong to the case of almost product structure, e.g., f-structure (i.e., ) and para-f-structure (i.e., ); such structures on singular distributions were considered in . Almost product structures on statistical manifolds were studied in [11,12].
(b) Let be a singular Riemannian foliation of , i.e., the leaves are smooth, connected, locally equidistant submanifolds of M. e.g., . Then is a singular distribution parameterized by the orthoprojector .
In this article, we generalize Bochner’s technique to a Riemannian manifold endowed with a singular (or regular) distribution and a statistical type connection, continue our study [13,14,15,16,17,18] and generalize some results of other authors in . Recall that the Bochner technique works for skew-symmetric tensors lying in the kernel of the Hodge Laplacian on a closed manifold: using maximum principles, one proves that such tensors are parallel, e.g., . Here d is the exterior differential operator, and is its adjoint operator for the inner product. The elliptic differential operator can be decomposed into two terms,
one is the Bochner Laplacian , and the second term (depends linearly on the Riemannian curvature tensor) is called the Weitzenböck curvature operator on -tensors S,e.g., .
Here is a local orthonormal frame on and is the -adjoint of the Levi–Civita connection ∇. Note that ℜ reduces to when evaluated on (0,1)-tensors, i.e., . According to the well-known formula for the action of the curvature tensor R on -tensors, for the formula from (2) has the form
or, in coordinates, . The Weitzenböck decomposition Formula (1) allows us to extend the Hodge Laplacian to arbitrary tensors and is important in the study of interactions between the geometry and topology of manifolds.
Our work has an Introduction section and eight subsequent sections, the References include 25 items. In Section 3, we generalize the notion of statistical structure for the case of distributions. In Section 2, Section 4 and Section 5, following an almost Lie algebroid construction (Section 8 with Appendix) we define the derivatives and , the modified divergence and their adjoint operators on tensors, and modified Laplacians on tensors and forms. In Section 6, making some assumptions about P (which are trivial when ), we define the curvature type operator of . In Section 7, we define the Weitzenböck type curvature operator on tensors, prove the Bochner–Weitzenböck type formula and obtain vanishing results. The assumptions that we use are reasonable, as illustrated by examples.
2. The Modified Covariant Derivative and Bracket
Here, we define the map satisfying Koszul conditions, see (48) in Section 8,
called a P-connection, which depends on P and a -tensor K (called contorsion tensor), but generally is not a linear connection on M. Set for (the P-gradient of f). In particular, for , we have the P-connection defined in  by
which plays, in our study, the same role as the Levi–Civita connection in metric-affine geometry. Using , we construct the P-derivative of -tensor S, where , as -tensor :
We use the standard notation . A tensor S is called P-parallel if .
A linear connection on a Riemannian manifold is metric if , e.g., ; in this case, , where is adjoint to with respect to g. This concept of metric-affine geometry can be applied for our P-connections. Recall that is metric, see .
The P-connection has a metric property, i.e., , if and only if the map , see (3), is skew-symmetric for any , that is .
If (8) holds for a P-connection (3), then the endomorphism P and the bracket given in (7) define an almost algebroid structure on .
This follows from Proposition 2, according to Definition 7 in Section 8. □
If (the Nijenhuis tensor of P) and (where and ∇ is the Levi–Civita connection of g), then the tensor (given in Proposition 2) is symmetric, thus the condition (8) holds.
3. The Statistical -Structure
A linear connection on a Riemannian manifold is called statistical if it is torsionless and tensor is symmetric in all its entries, e.g., [6,9]. Such a pair is called a statistical structure on M. In this case,
equivalently, the statistical cubic form is symmetric. We generalize this concept for singular distributions.
A P-connection on will be called statistical if the statistical cubic form is symmetric, or, equivalently, (10) holds. In this case, the pair is called a statistical P-structure on M.
If is a statistical P-connection for g then the (3,0)-tensor is symmetric in all its entries, i.e., the following Codazzi type condition holds:
The theory of Codazzi tensors is well described in . By (6), (10) and the property , we have , thus all three terms in (11) are equal. □
Since for the Levi–Civita connection ∇, condition (11) does not impose restrictions on P and it is equivalent to the property “the cubic form A is symmetric".
By (9) and (10), the P-bracket of a statistical P-structure does not depend on K:
If is statistical then , see (4), has the same P-bracket and . Proposition 2 yields the following result for a statistical P-structure.
For a statistical P-structure, condition , see (8), is equivalent to
We can put and reduce (8) to a simpler view (13). □
The notion of conjugate connection is important for statistical manifolds, see [9,20].
For a P-connection on , its conjugate P-connection is defined by the following equality:
One may show that holds in general, thus, for a statistical P-connection the conjugate connection is given by . In turn, the statistical P-connection is conjugate to . Note that .
For a conjugate statistical P-connection , we can define the P-bracket by and the tensor . By (10), we have
From Proposition 3, using Remark 1, we obtain the following corollaries.
The pairs and are simultaneously statistical P-structures on M.
A statistical P-structure on and its conjugate simultaneously define almost algebroid structures (see definition in Section 8) on .
To simplify the calculations, for the rest of this article we will restrict ourselves to statistical P-structures, see (10), and to use the concept of almost Lie algebroid, assume (13).
The following proposition extends result for regular case, and in .
Let be a complete non-compact Riemannian manifold endowed with a vector field X such that (or , where such that conditions (17) and hold. Then, .
Let be the -form in M given by , i.e., the contraction of the volume form in the direction of . If is an orthonormal frame on an open set with coframe , then
Since the -forms are orthonormal in , we get . Thus, and
see (18). There exists a sequence of domains on M such that , and , see . Then
But since on M, it follows that on M. □
We call the P-Laplacian for functions. Using (3), we have
that generalizes Lemma 6.1 in  for regular case, .
Consider the following system of singular distributions on a smooth manifold M: , , etc. The distribution is said to be bracket-generating of the step if , e.g., . Note that integrable distributions, i.e., , are not bracket-generating. The condition means that is constant along the (integral curves of) ; moreover, if is bracket-generating then on M.
The next theorem extends the well-known classical result on subharmonic functions and generalizes Theorem 1 in  (see also  for and ).
Let conditions (17) hold for a statistical P-connection , and let satisfy either or . Suppose that any of the following conditions hold:
(a) is closed;
(b) is complete non-compact, and belong to .
Then, ; moreover, if is bracket-generating, then .
(a) Using Theorem 2, we get . By the equality with ,
and again Theorem 2 with , we get , hence .
(b) By Proposition 8 with and condition , we get . Using (31) with , Proposition 8 with and condition , we get , hence . If the distribution is bracket-generating, then using Chow’s theorem  completes the proof for both cases. □
6. The Modified Curvature Tensor
Define the second P-derivative of an -tensor S as the -tensor
5. Since , see 1., the first equality follows. For the second one, we use 2:
thus, the claim follows from the equality . □
Similarly, we define the P-curvature tensor of the conjugate P-connection ,
The following curvature type tensor (depending on P only) has been introduced in :
Since we assume then holds. By the above,
and when (35) holds. The Ricci tensor of was defined in  by
For a statistical P-structure, we have
Thus, is symmetric if and only if is symmetric.
Using symmetry of K, we have
From the above the claim follows. □
The endomorphism P of induces endomorphisms and its adjoint of :
see . The curvature tensor can be seen as a self-adjoint linear operator on the space of bivectors, called the curvature operator, e.g., [7,19]. Similarly, we consider as a linear operator or as a corresponding bilinear form on . For this, using skew-symmetry of for a statistical P-connection, define a linear operator on by
and observe (symmetry). Put and , i.e.,
The above generalizes , having the properties, see ,
Using known properties of and property 4. of , we have
Note that if then on is not self-adjoint:
7. The Weitzenb öck Type Curvature Operator
Here, we use the P-connection to introduce the central concept of the paper: the Weitzenböck type curvature operator on tensors. We generalize the Weitzenböck curvature operator (2), (see also  for statistical manifolds when , and  for distributions when ) for the case of distributions with statistical structure.
Define the P-Weitzenböck curvature operator on -tensors S over by
The operators and are defined similarly using P-connections and .
For a differential form , the is skew-symmetric. Note that reduces to when evaluated on (0,1)-tensors, i.e., . For using (34), the formula from (37) reads as
or, in coordinates, .
The following lemma represents using and K.
For a statistical P-structure, let (21) hold. Then we have
Lemma 6 allows us to rewrite the operator (37). The following result generalizes Proposition 8 in .
If S is a -tensor on , then
In particular, if P is self-adjoint, then is self-adjoint too.
We follow similar arguments as in the proof of Lemma 9.3.3 in :
Thus, the first claim follows. Since is self-adjoint, there is a local orthonormal base of such that . Using this base, for any -tensors and , we get
and, similarly, again using ,
Thus, the second claim follows. □
The following result generalizes Corollary 9.3.4 in  and Proposition 10 in .
Let be a statistical P-structure on a manifold M.
(a) If for any -tensor S, then .
(b) Moreover, if for any -tensor S, where , then
where a constant C depends only on the type of S.
Using (45) and a local orthonormal base of such that , we get
By conditions, for all , thus, , and the first claim follows. There is a constant depending only on the type of the tensor and such that , see Corollary 9.3.4 in . By conditions, for all . The above yields – thus, the second claim. □
Let (21) be satisfied for a statistical P-structure on a closed manifold M and for any k-form ω. Then any P-harmonic k-form on M is -parallel.
By conditions and Proposition 13(a), . By (41), since , we get . By Proposition 5, we have . □
The following result extends Theorem 3 with and in .
Let (21) be satisfied for a statistical P-connection on a complete non-compact and for some and all . Suppose that for any 1-form ω, where C is defined in Proposition 13(b). If for a P-harmonic 1-form ω, then .
By conditions, Remark 3 and Proposition 13(b),
By (43) with , since , see (29), we get . By Proposition 8 with and , we get . Applying Theorem 3(b), we get . □
Notice that, if in Theorems 5 and 6 is bracket-generating, then on M.
8. Supplement On the Almost Lie Algebroid Structure
Here, for the convenience of a reader, we briefly recall the construction of an almost Lie algebroid, following Section 2 in  (see also [15,16]). Lie algebroids (and Lie groupoids) constitute an active field of research in differential geometry. Roughly speaking, an (almost) Lie algebroid is a structure, where one replaces the tangent bundle of a manifold M with a new smooth vector bundle of rank k over M (i.e., a smooth fiber bundle with fiber ) with similar properties. Lie groupoids are related to Lie algebroids similarly as Lie groups are related to Lie algebras, see . Lie algebroids deal with integrable distributions (foliations). Almost Lie algebroids are closely related to singular distributions, e.g., [13,14].
An anchor on is a morphism of vector bundles. A skew-symmetric bracket on is a map such that
for all and . The anchor and the skew-symmetric bracket give an almost Lie algebroid structure on .
Note that axiom (46), third formula, is equivalent to vanishing of the following operator:
There is a bijection between almost Lie algebroids on and the exterior differentials of the exterior algebra ,17]; here is the set of k-forms over . The exterior differential , corresponding to the almost Lie algebroid structure , is given by
where and for . For , we have , where and . Recall that a skew-symmetric bracket defines uniquely an exterior differential on , and it gives rise to
an almost algebroid if and only if for ;
a Lie algebroid if and only if and for and .
A -connection on is a map satisfying Koszul conditions
For a -connection on , they define the torsion and the curvature by standard formulas
The main contribution of this paper is the further development of Bochner’s technique for a regular or singular distribution parameterized by a smooth endomorphism P of the tangent bundle of a Riemannian manifold with linear connection. In particular, the main results of this paper, Theorems 1–6 are proved. We introduce the concept of a statistical P-structure, i.e., a pair of a metric g and P-connection on M with a totally symmetric contorsion tensor K, see (10), and assume (13) for P to use the concept of almost Lie algebroids. To generalize some geometrical analysis tools for distributions, we assume the additional conditions (21) and (35) for tensors P and K. We introduce and study a Weitzenböck type curvature operator on tensors and prove vanishing theorems on the null space of the Hodge type Laplacian on a distribution with a statistical type connection.
We delegate the following for further study: (a) generalize some constructions in the paper, e.g., statistical P-structures, divergence results, to more general almost algebroids or Lie algebroids; (b) use less restrictive conditions on K; (c) find more applications in geometry and physics.
Investigation, P.P., V.R. and S.S. All authors contributed equally in writing this article. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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