Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Divergence Identity for the scalar curvature and Rigidity of Codazzi Tensors

Xu Cheng, Andrés Lipa, Detang Zhou

TL;DR

The paper develops a divergence identity for a locally defined vector field $X$ on an $n$-manifold, linking $\mathrm{div}(X)$ to the scalar curvature via $\mathrm{div}(X) = \tfrac{1}{2} S_M - \Psi$ where $\Psi$ depends on connection coefficients. Using this identity together with a detailed analysis of the eigenvalues of Codazzi symmetric $(0,2)$-tensors and the invariants $\sigma_k(A)$, the authors obtain a new proof of the Tang–Yan rigidity theorem and extend rigidity results to broader settings, including partial constancy of symmetric functions and vanishing determinant cases. They apply the results to isoparametric rigidity of hypersurfaces in spheres and rigidity of the Ricci tensor under harmonic curvature, illustrating the method's versatility. The work unifies divergence-based techniques with eigenvalue polynomial machinery to derive constancy of eigenvalues and curvature conditions, with potential impact on geometric rigidity problems and isoparametric theory.

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.

Divergence Identity for the scalar curvature and Rigidity of Codazzi Tensors

TL;DR

The paper develops a divergence identity for a locally defined vector field on an -manifold, linking to the scalar curvature via where depends on connection coefficients. Using this identity together with a detailed analysis of the eigenvalues of Codazzi symmetric -tensors and the invariants , the authors obtain a new proof of the Tang–Yan rigidity theorem and extend rigidity results to broader settings, including partial constancy of symmetric functions and vanishing determinant cases. They apply the results to isoparametric rigidity of hypersurfaces in spheres and rigidity of the Ricci tensor under harmonic curvature, illustrating the method's versatility. The work unifies divergence-based techniques with eigenvalue polynomial machinery to derive constancy of eigenvalues and curvature conditions, with potential impact on geometric rigidity problems and isoparametric theory.

Abstract

We introduce a local vector field on an -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 -dimensional manifold with nonnegative scalar curvature, a smooth Codazzi symmetric tensor whose trace invariants up to order 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.
Paper Structure (10 sections, 27 theorems, 169 equations, 2 figures)

This paper contains 10 sections, 27 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Vector field
  3. Identities for symmetric 2-tensors and connection coefficients
  4. Expressions for $\Psi$ and $\langle X, \nabla \sigma_n\rangle$
  5. Proof of Theorem \ref{['T-Y']}
  6. Vanishing determinant case
  7. Applications to shape operator and Ricci tensor
  8. Hypersurfaces in the sphere
  9. Ricci tensor
  10. Appendix

Key Result

Theorem 1.1

Let $\{e_1,\dots,e_n\}$ be an orthonormal frame on an open subset $D$ of an $n$-dimensional Riemannian manifold $M^n$. Let $X$ be the vector field on $D$ defined by Then the divergence of $X$ satisfies where $S_M$ denotes the scalar curvature of $M^n$, and

Figures (2)

  • Figure 1: An example illustrating the case of equality in \ref{['ine:odd-1']}
  • Figure 2: An example illustrating the case of equality in \ref{['ine:even-1']}

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2: Almeida--Brito Almeida-Brito90
  • Theorem 1.3: Tang--Yan Tang-Yan23
  • Corollary 1.4: Tang--Yan Tang-Yan23
  • Conjecture 1.5: Chern
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.1
  • ...and 53 more