Divergence identity for the scalar curvature and rigidity of Codazzi tensors

Abstract: In this joint work with Xu Cheng and Detang Zhou we introduce a local vector field on an $n$-dimensional Riemannian manifold, defined as the sum of the covariant derivatives of a local orthonormal frame, and derive an explicit identity for its divergence, decomposed into a scalar curvature term and an auxiliary term involving connection coefficients. This result is applied to rigidity problems for Codazzi symmetric tensors. In particular, we give a new proof of a Tang-Yan theorem, which states that on a closed $n$-dimensional manifold with nonnegative scalar curvature, a smooth Codazzi symmetric tensor whose trace invariants up to order $n − 1$ are constant must have constant eigenvalues. We also obtain further rigidity results under assumptions on elementary symmetric functions of the eigenvalues, with applications to the isoparametric rigidity of closed hypersurfaces in the unit sphere.

Portuguesedifferential geometry

Audience: researchers in the topic

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Geometry Webinar AmSur /AmSul

Series comments: The Geometry Webinar AmSur/AmSul is promoted by differential geometry groups from Universities in Argentina and Brazil. The webinar happens weekly on Fridays at 14h00 (GMT-3) via Google Meet. Talks will be in Spanish, Portuguese or English. This virtual seminar aims to establish the contact with several research groups and mathematicians from Latin America. Everybody is invited to participate writing to the contact e-mail: geodif@unicamp.br

For each talk, the Google Meet link will be sent.

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