Weyl-Schouten Theorem for symmetric spaces

TL;DR

This paper proves that a Riemannian manifold with a Weyl tensor proportional to that of a symmetric space without constant curvature factors is conformally equivalent to that symmetric space, extending the Weyl-Schouten theorem.

Contribution

It generalizes the Weyl-Schouten theorem to a broader class of symmetric spaces with specific de Rham decompositions.

Findings

Manifolds with Weyl tensor proportional to that of N are conformally equivalent to N.

The result applies to symmetric spaces of dimension greater than 5 without constant curvature factors.

Extension of the classical Weyl-Schouten theorem to new geometric settings.

Abstract

Let N be a symmetric space of dimension n > 5 whose de Rham decomposition contains no factors of constant curvature and let W be the Weyl tensor of N at some point. We prove that a Riemannian manifold whose Weyl tensor at every point is a positive multiple of W is conformally equivalent to N (the case N = R^n is the Weyl-Schouten Theorem).