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VOL. 1 | 2018 Chapter 16. The Riemann extension of an affine Osserman connection on 3-dimensional manifold
Abdoul Salam DIALLO

Editor(s) Hamet SEYDI, Gane Samb LO, Aboubakary DIAKHABY

## Abstract

The Riemannian extension of torsion free affine manifolds $(M, \nabla)$ is an important method to produce pseudo-Riemannian manifolds. It is known that, if the manifold $(M, \nabla)$ is a torsion-free affine two-dimensional manifold with skew symmetric tensor Ricci, then $(M, \nabla)$ is affine Osserman manifold. In higher dimensions the skew symmetric of the tensor Ricci is a necessary but not sufficient condition for a affine connection to be Osserman. In this paper we construct affine Osserman connection with Ricci flat but not flat and example of Osserman pseudo-Riemannian metric of signature $(3,3)$ is exhibited.

## Information

Published: 1 January 2018
First available in Project Euclid: 26 September 2019

Digital Object Identifier: 10.16929/sbs/2018.100-03-04

Subjects:
Primary: 53B05 , 53B20 , 53C30

Keywords: affine connection , Jacobi operator , Osserman manifold , Riemann extension