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

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

2009-10-09

arxiv.org/abs/0910.1645v1

Mathematics

Differential Geometry

25 pages

Scientific paper

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.

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