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Open AccessArticle

The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms

1
Department of Mathematics, University of Haifa, Haifa 3498838, Israel
2
Department of Mathematics, Russian Institute for Scientific and Technical Information of the Russian Academy of Sciences, Moscow 125190, Russia
3
Department of Data Analysis and Financial Technologies, Finance University, Moscow 125993, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 83; https://doi.org/10.3390/math8010083
Received: 26 November 2019 / Revised: 23 December 2019 / Accepted: 24 December 2019 / Published: 3 January 2020
(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)
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. View Full-Text
Keywords: Riemannian manifold; Bourguignon Laplacian; symmetric bilinear form; harmonic; curvature; spectral theory; vanishing theorem Riemannian manifold; Bourguignon Laplacian; symmetric bilinear form; harmonic; curvature; spectral theory; vanishing theorem
MDPI and ACS Style

Rovenski, V.; Stepanov, S.; Tsyganok, I. The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms. Mathematics 2020, 8, 83.

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