Affine projective Osserman structures

Abstract

By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank 1 symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank 1 Riemannian symmetric space with the modified Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection in question is not the Levi-Civita connection of any metric. If the dimension is odd, we use methods of algebraic topology to show the Jacobi operator of an affine projective Osserman manifold has only one non-zero eigenvalue and that eigenvalue is real.1.2. Osserman geometry in the pseudo-Riemannian geometry. Suppose that M = (M, g) is a pseudo-Riemannian manifold of signature (p, q) for p > 0 and q > 0. The pseudo-sphere bundles are defined by setting:One says that (M, g) is spacelike (resp. timelike) Osserman if the eigenvalues of J are constant on S ± (M, g). The situation is rather different here as the Jacobi 2000 Mathematics Subject Classification. 53C50, 53C44. Key words and phrases. affine Osserman, affine projective Osserman, spacelike projective Osserman, timelike projective Osserman. 1 x 1 := cos θe 1 + sin θe 3 ,x 2 := − sin θe 1 + cos θe 3 , x 3 := Jx 1 = cos θe 2 + sin θe 4 , x 4 := Jx 2 = − sin θe 2 + cos θe 4 .Let ⋆ be a coefficient which we do not need to specify. We then haveJ (x 1 )x 3 = (λ 0 + 3λ 1 )x 3 + ⋆e 1 = ⋆x 1 + ⋆x 2 + (λ 0 + 3λ 1 )x 3 , J (x 1 )x 4 = λ 0 x 4 + ⋆e 1 = ⋆x 1 + ⋆x 2 + λ 0 x 4 .The matrix of J (x 1 ) on this 4-dimensional subspace is therefore given by

Projective affine Osserman curvature models

“…If m ≡ 1(2), then there is one eigenbundle. One uses an argument using characteristic classes (see [27]) to rule out the case that the eigenbundle relates to a complex eigenvalue and conclude that the eigenvalue λ is real; then an example may be obtained by taking a space of constant sectional curvature λ or, if positive, by using a rescaled version of the manifold given in Example 1.6. This completes the proof of Theorem 1.11 if m ≡ 1 (2).…”

“…One may then verify [27] that this is projective affine Osserman. The entries in the curvature tensor are constant so this manifold is affine curvature homogeneous.…”

“…In the following result, we shall omit the λ's if there are no real eigenvalues and the µ's if there are no complex eigenvalues. We refer to [27] for proof of: Theorem 1.10. Let (V, A) be a projective affine Osserman curvature model.…”

“…To illustrate this, we present two examples of projective affine Osserman manifolds that do not arise from an underlying Riemannian structure and refer to [27] for other examples: Example 1.6. Let {e 1 , ..., e m } be the standard basis for R m .…”