Osserman manifolds and Weyl-Schouten Theorem for rank-one symmetric spaces

TL;DR

This paper proves the Osserman and Conformal Osserman Conjectures, establishing that such manifolds are either flat or rank-one symmetric, and shows conformal equivalence to rank-one symmetric spaces under certain conditions.

Contribution

It confirms the Osserman and Conformal Osserman Conjectures, advancing understanding of curvature tensor properties in Riemannian geometry.

Findings

Osserman Conjecture is proven under specific algebraic assumptions.

Conformal equivalence to rank-one symmetric spaces is established.

Results hold in the context of Weyl tensor analysis.

Abstract

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in $\mathbb{R}^16$. As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space, is conformally equivalent to it.