The spectral geometry of the Weyl conformal tensor

Abstract

Abstract. We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.

Complex Osserman Kähler manifolds in dimension four

“…Thus there are two J terms A(J * , * , J * , * ) which, modulo the Kähler identity, are of the form A(Jx 1 , x 2 , Jx 1 , x 2 ) or A(Jx 1 , x 1 , Jx 2 , x 2 ) which have already been discussed. This proves Assertion (2).…”

“…Complex lines {π 1 , ..., π k } in CP(H) will be said to form a µconfiguration if π i ⊂ E µ (π i ) and if π i ⊥ π j for i = j; this then implies π j ⊂ E λ (π i ) for i = j. 1 Lemma 4.2. Given H as above, there exist complex lines {π 1 , ..., π n 2 } which form a µ-configuration.…”

“…Since λ = 0 and µ = 0, H 1 does not have constant holomorphic sectional curvature. Consequently we may proceed inductively to construct a µ-configuration {π 2 , ..., π 1 2 n } for H 1 ; {π 1 , ..., π 1 2 n } is then a µ-configuration for H. If n = 4, then Theorem 1.8 follows from the following result: Lemma 4.3. Let n = 4 and let {π 1 , π 2 } be a µ-configuration for H. Fix x 1 ∈ S(π 1 ).…”

“…(1) If R(x, y, y, x) = c for any orthonormal set {x, y}, then R is said to have constant sectional curvature. (2) The Jacobi operator is defined by J (x) : y → R(y, x)x.…”

“…There are related problems defined by other natural operators. The conformal Jacobi operator has been investigated [1,2,18], the skew-symmetric curvature operator has been investigated [12,13,16], and the higher order Jacobi operator has been investigated [10]; we refer to [8,11] for further details.…”