OSSERMAN MANIFOLDS AND CLIFFORD STRUCTURES Y. NIKOLAYEVSKY

Abstract. A Riemannian manifold M is called Osserman if the eigenvalues of the Jacobi operator RX = R(X, ·)X are constant on the unit tangent bundle. R. Osserman conjectured that any such manifold is two-point homogeneous. A Cliff(ν)-structure on a Riemannian manifold M n is a collection of ν almost Hermitian structures J1,..., Jν, subject to Hurwitz relations JiJj + JjJi = −2δijid, such that RXY = λ0(‖X‖2 Y − 〈X, Y 〉X) + P i λi〈JiX, Y 〉JiX for some constants λ0, λ1,..., λν. A manifold with a Cliff(ν)-structure is Osserman. The standard approach to the Osserman Conjecture is (1) to show that any Osserman manifold admits a Cliff(ν)structure and (2) to classify Riemannian manifolds with a Cliff(ν)-structure. We prove that (1) if the Jacobi operator of an Osserman manifold M of dimension not divisible by 8 has a simple eigenvalue λ, then M admits an almost Hermitian structure J such that RXJX = λJX for any unit vector X; (2) A manifold M n with a Cliff(ν)-structure is two-point homogeneous, provided n ̸ = 2, 4, 8 and (n, ν) ̸ = (16, 8). 1