scholarly journals On Singular Distributions With Statistical Structure

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1825
Author(s):  
Paul Popescu ◽  
Vladimir Rovenski ◽  
Sergey Stepanov
Keyword(s):  
Riemannian Manifold ◽  
Null Space ◽  
Curvature Operator ◽  
Lie Algebroid ◽  
Central Concept ◽  
Statistical Structure ◽  
Type Formula ◽  
Regular Distribution

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.

scholarly journals The Weitzenböck Type Curvature Operator for Singular Distributions

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 365
Author(s):  
Paul Popescu ◽  
Vladimir Rovenski ◽  
Sergey Stepanov
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Tangent Bundle ◽  
Null Space ◽  
Curvature Operator ◽  
Vanishing Theorems ◽  
Lie Algebroid ◽  
Decomposition Formula ◽  
Adjoint Operators ◽  
Type Decomposition

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

Eigenvalues of the Curvature Operator for Certain Homogeneous Manifolds

1990 ◽  
Vol 42 (6) ◽  
pp. 981-999
Author(s):  
J. E. D'Atri ◽  
I. Dotti Miatello
Keyword(s):  
Riemannian Manifold ◽  
Tangent Space ◽  
Orthonormal Basis ◽  
Riemann Tensor ◽  
Inner Product ◽  
Curvature Operator ◽  
Curvature Function ◽  
Homogeneous Manifolds ◽  
Exterior Power ◽  
Image Position

Given a Riemannian manifold M, the Riemann tensor R induces the curvature operator on the exterior power of the tangent space, defined by the formula where the inner product is defined by From the symmetries of R, it follows that ρ is self-adjoint and so has only real eigenvalues. R also induces the sectional curvature function K on 2-planes in is an orthonormal basis of the 2-plane π.

The Proof of the Feynman–Kac Formula for Heat Equation on a Compact Riemannian Manifold

2003 ◽  
Vol 06 (02) ◽  
pp. 311-320 ◽  
Author(s):  
Oleg O. Obrezkov
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Compact Riemannian Manifold ◽  
Type Formula ◽  
Chernoff Theorem ◽  
Common Lines ◽  
Full Proof

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

scholarly journals A new class of almost complex structures on tangent bundle of a Riemannian manifold

2018 ◽  
Vol 26 (2) ◽  
pp. 137-145
Author(s):  
Amir Baghban ◽  
Esmaeil Abedi
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Integrability Condition ◽  
Complex Structure ◽  
Riemannian Metric ◽  
Curvature Operator ◽  
Complex Structures ◽  
New Class ◽  
Almost Complex Structures ◽  
Almost Complex

AbstractIn this paper, the standard almost complex structure on the tangent bunle of a Riemannian manifold will be generalized. We will generalize the standard one to the new ones such that the induced (0, 2)-tensor on the tangent bundle using these structures and Liouville 1-form will be a Riemannian metric. Moreover, under the integrability condition, the curvature operator of the base manifold will be classified.

Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

2004 ◽  
Vol 70 (2) ◽  
pp. 301-319 ◽  
Author(s):  
Y. Nikolayevsky
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Curvature Operator ◽  
Specific Function ◽  
3 Dimensional ◽  
Space Of Constant Curvature

A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

RENORMALIZED LAPLACIANS ON A CLASS OF HILBERT MANIFOLDS AND A BOCHNER–WEITZENBÖCK TYPE FORMULA FOR CURRENT GROUPS

2000 ◽  
Vol 03 (01) ◽  
pp. 53-98 ◽  
Author(s):  
MARC ARNAUDON ◽  
YANA BELOPOLSKAYA ◽  
SYLVIE PAYCHA
Keyword(s):  
Hilbert Space ◽  
Riemannian Manifold ◽  
Differential Operators ◽  
Sobolev Class ◽  
Hilbert Manifold ◽  
One Dimensional ◽  
Type Formula ◽  
Pseudo Differential Operators ◽  
Underlying Manifold ◽  
Class Of Functions

We define renormalized Laplacians and investigate their properties for a class of C2 functions on Hilbert manifolds modeled on a Hilbert space Hs(M, V) of sections of Sobolev class Hs of some vector bundle V based on a closed Riemannian manifold M, and such that the transition maps of the Hilbert manifold are pseudo-differential operators acting on smooth sections of the bundle V. Among these manifolds we find current groups Hs(M, G), s> dim M/2 where M is a closed manifold and G a Lie group. Weighted Laplacians are renormalized Laplacians which coincide with ordinary Laplacians when the underlying manifold M reduces to a point. We prove a Bochner–Weitzenböck type formula for weighted Laplacians and we point out how in some cases it can reduce to a relation of the type [Δ,∇]f= Ricci (∇f, ·) for a class of functions on certain current groups. Another type of renormalized Laplacian we define are Lévy-type Laplacians which coincide with Lévy Laplacians when the underlying manifold M is one-dimensional. We describe them as limit generators for a one-parameter family of regularized Brownian motions.

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