Conformally Osserman Manifolds and Conformally Complex Space Forms

Abstract

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.

Conformally Osserman manifolds

“…Theorem 3 answers, with three exceptions, the conjecture made in [BGNSi] (for conformally Osserman manifolds of dimension n > 6 not divisible by 4, this conjecture is proved in [BG1,Theorem 1.4]).…”

Weyl-Schouten Theorem for symmetric spaces

“…In the earlier papers, the Theorem was established for M 0 = CP m , m ≥ 4 (and for its noncompact dual) [BG1], for rank-one symmetric spaces of dimension n > 4 [ N3,Theorem 2], and for simple groups with a bi-invariant metric [N1]. The dimension restriction in the Theorem excludes only the spaces CP 2 and SU (3)/SO(3) and their duals.…”

“…In this section we prove Proposition 3 for the complex and the quaternionic projective spaces, and also the fact that Φ = 0 for the Cayley projective plane (the fact that K = 0 for M 0 = OP 2 then follows from Lemma 6(3)). Note that the proof of a statement equivalent to Proposition 3 for rank one compact symmetric spaces is contained "in disguise" in [N3, N2] under more general assumptions; for the complex projective space, see [BG1]. For completeness, we give a direct proof here.…”

Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

“…The proof of the Osserman Conjecture for manifolds of dimension not divisible by 4 was given in [Chi], before the conjecture itself was published. The Conformal Osserman Conjecture for manifolds of dimension n > 6, not divisible by 4, is proved in [BG1], for manifolds with the structure of a warped product, both conjectures are proved in [BGV].…”