Higher order Codazzi tensors on conformally flat spaces
type: Item
instance of: scholarly article
title: Higher order Codazzi tensors on conformally flat spaces
zbMATH Open document ID: 0919.53006
author: Udo Simon
author: Huili Liu
author: Changping Wang
published in: Beiträge zur Algebra und Geometrie
publication date: 1999-09-07T00:00:00Z
full work available at URL: https://eudml.org/doc/228646
full work available at URL: no.2
review text: A symmetric \((0,m)\)-tensor field \(\phi\) is called a Codazzi tensor field with respect to an affine connection \(\nabla\) if and only if the covariant derivative of \(\phi\) with respect to \(\nabla \) is totally symmetric. Codazzi tensors naturally appear in the study of various classes of submanifolds. For example, parallel submanifolds of real and complex space forms are examples of manifolds admitting a Codazzi tensor of type \((0,2)\). Also, in affine differential geometry, the affine metric \(h\) can be seen as a Codazzi tensor of type \((0,2)\) with respect to the induced affine connection. In that setting, affine spheres can be characterized by the condition that the cubic form is a Codazzi tensor of type \((0,3)\) with respect to the Levi-Civita connection of the affine metric \(h\). In this paper, the authors give a short survey about the different domains in which such Codazzi tensors occur. In the case that the connection is the Levi-Civita connection of a semi-Riemannian metric \(g\), they also obtain a formula for the Laplacian of the length of \(\phi\). A formula, which under some additional assumptions (tracelessness of \(\phi\), conformally flat metric) becomes surprisingly nice. In the final section, some global consequences of this formula are presented, including some pinching theorems involving the Ricci curvature.
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1208008
zbMATH Keywords: Codazzi tensors
zbMATH Keywords: affine differential geometry
zbMATH Keywords: Laplacian
zbMATH Keywords: pinching
reviewed by: Luc Vrancken
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1208008