Second order local affine invariants
type: Item
instance of: scholarly article
title: Second order local affine invariants
zbMATH Open document ID: 0919.53003
author: Peter B. Gilkey
author: Neda Bokan
author: Udo Simon
published in: Russian Mathematics
publication date: 1996-05-12T00:00:00Z
review text: Let \(g\) be a Riemannian metric and \(\nabla\), \(\nabla^*\) two torsion free Ricci symmetric connections. A triple \((\nabla, g, \nabla^*)\) is said to be conjugate if for all vector fields \(U, V, W\) the equality \[ U_g(V,W) = g(\nabla_U V,W) + g(V,\nabla^*_UW) \] holds. For a conjugate triple, the tensors \(\nabla g\) and \(\nabla^*g\) satisfy the classical Codazzi equations. Also, in this case, the Levi Civita connection of the metric \(g\) is given by \(\nabla^g ={1\over 2}(\nabla + \nabla^*)\). Next, the authors define the tensor \(C = \nabla - \nabla^g\) and a special transformation of \(C\) associated with a conformal transformation of \(g\). This transformation has a number of properties permitting construction of local invariants. The authors describe and study such invariants. They also study various special cases in which the assumption that these invariants have specific values forces the underlying manifold to be of special type, such as a sphere, a hyperovaloid, etc.
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 857509
zbMATH Keywords: local invariants
zbMATH Keywords: Codazzi invariants
zbMATH Keywords: Codazzi equations
reviewed by: Vladimir I. Oliker
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 857509