Conformally Osserman manifolds and self-duality in Riemannian geometry

2005/04/25 by Novica Blazic, Peter Gilkey, Blazic, Novica +1
Mathematics · Physics and Astronomy · #53B20 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP #msc:53B20

paper · pdf · doi:10.48550/arxiv.math/0504498

arxiv created 2005/04/25 · arxiv updated 2009/12/01

Abstract

We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the curvature tensor of (M,g) is given by a quaternionic structure, at least pointwise.

Related

• Conformally Osserman manifolds
• The spectral geometry of the Weyl conformal tensor
• Osserman manifolds and Weyl-Schouten Theorem for rank-one symmetric spaces
• Conformally flat Einstein-like 4-manifolds and conformally flat Riemannian 4-manifolds all of whose Jacobi operators have parallel eigenspaces along every geodesic
• Osserman Conjecture in dimension n ≠ 8, 16