The bourguignon laplacian and harmonic symmetric bilinear forms

Vladimir Rovenski; Sergey Stepanov; Irina Tsyganok

doi:10.3390/math8010083

The bourguignon laplacian and harmonic symmetric bilinear forms

Vladimir Rovenski
, Sergey Stepanov
, Irina Tsyganok

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.

Original language	English
Article number	83
Journal	Mathematics
Volume	8
Issue number	1
DOIs	https://doi.org/10.3390/math8010083
State	Published - 1 Jan 2020

Bibliographical note

Publisher Copyright:
© 2020 by the authors.

Keywords

Bourguignon Laplacian
Curvature
Harmonic
Riemannian manifold
Spectral theory
Symmetric bilinear form
Vanishing theorem

ASJC Scopus subject areas

Computer Science (miscellaneous)
General Mathematics
Engineering (miscellaneous)

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10.3390/math8010083

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AU - Stepanov, Sergey

AU - Tsyganok, Irina

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N2 - In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.

AB - In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.

KW - Bourguignon Laplacian

KW - Curvature

KW - Harmonic

KW - Riemannian manifold

KW - Spectral theory

KW - Symmetric bilinear form

KW - Vanishing theorem

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ER -

The bourguignon laplacian and harmonic symmetric bilinear forms

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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