Divergence Identity for the scalar curvature and Rigidity of Codazzi Tensors

TL;DR

This paper introduces a divergence identity related to scalar curvature on Riemannian manifolds and applies it to prove new rigidity results for Codazzi tensors, including a new proof of a known theorem and applications to hypersurface geometry.

Contribution

It provides a new divergence identity and uses it to establish rigidity results for Codazzi tensors, including a novel proof of Tang-Yan theorem and applications to isoparametric hypersurfaces.

Findings

New divergence identity for scalar curvature and connection coefficients.

Rigidity results for Codazzi symmetric tensors with constant trace invariants.

Applications to the geometry of hypersurfaces in spheres.

Abstract

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.