arXiv subjects

S. E. Stepanov

Publications and source records attributed to S. E. Stepanov.

6 recordsLinked to original sources

arXiv2023

Relationship between sectional curvature and null spaces of Lichnerowicz-type Laplacians and their smallest eigenvalues

The first variant of this article contained a fatal error. Therefore, we publish second version our paper. In the present paper, we prove that the curvature operator of the second kind of a Riemannian manifold is strictly positive if its sectional curvature is strictly positive and the Ricci curvature suitably pinched. In addition, we prove several vanishing theorems for null spaces of the Lichnerowicz, Sampson, and Hodge-de Rham Laplacians and find estimates for their lowest eigenvalues on compact (without boundary) Riemannian manifolds with sectional pinched curvature.

math.DG↗

arXiv2018

On higher order Codazzi tensors on complete Riemannian manifolds

We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions. In conclusion, we show applications of this method for global geometry of a complete locally conformally flat Riemannian manifold with constant scalar curvature because its Ricci tensor is a Codazzi tensor and for global geometry of a complete hypersurface in a standard sphere because its second fundamental form is also a Codazzi tensor.

math.DG↗

arXiv2015

The Killing tensors on an $n$ -dimensional manifold with $SL(n, R)$-structure

In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $ SL(n, R) $ -structure and discuss possible applications of obtained results in Riemannian geometry.

math.DG↗

arXiv2014

A remark on the Laplacian operator which acts on symmetric tensors

More than forty years ago J. H. Samson has defined the Laplacian $Δ_{sym}$ acting on the space of symmetric covariant $p$-tensors on an $n$-dimensional Riemannian manifold $(M, g)$. This operator is an analogue of the well known Hodge-de Rham Laplacian $Δ$ which acts on the space of exterior differential $p$-forms ($1 \le p \le n$) on $(M, g)$. In the present paper we will prove that for $n > p = 1$ the operator $Δ_{sym}$ is the Yano rough Laplacian and show its spectrum properties on a compact Riemannian manifold.

math.DG↗

arXiv2014

On a Laplacian which acts on symmetric tensors

In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenböck decomposition and further of estimating its lowest eigenvalue.

math.DG↗

arXiv2013

Betti and Tachibana numbers

We present a rough classification of differential forms on a Riemannian manifold, we consider definitions and properties of conformal Killing forms on a compact Riemannian manifold and define Tachibana numbers as an analog of the well known Betti numbers. We state the conditions that characterize these numbers. In the last section we show connections between the Betti and Tachibana numbers.

math.DG↗