Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues

Abstract

Conformal Osserman four-dimensional manifolds are studied with special attention to the construction of new examples showing that the algebraic structure of any such curvature tensor can be realized at the differentiable level. As a consequence one gets examples of anti-self-dual manifolds whose anti-self-dual curvature operator has complex eigenvalues.

Examples of signature (2, 2) manifolds with commuting curvature operators

“…Assertions (1) and (2) of Theorem 1.2 follow from work of [5]. It is immediate that (3a) implies (3b).…”

“…One of the crucial features we shall exploit is that ρ, J (x), and R(x) are polynomial in the jets of g 34 . One has by [5] that the non-zero components of the curvature tensor are, after adjusting for a difference in the sign convention used therein, given by: One can now use the metric to raise indices and compute J , R, and ρ.…”

“…We have not attempted to obtain the most general possible classification results for Walker signature (2, 2) manifolds as our experience in similar related problems is that these tend to be excessively technical; for example, there is as yet no classification of Einstein Walker signature (2, 2) manifolds and there is as yet no classification of anti-self-dual Walker signature (2, 2) manifolds. As the family we shall examine has been studied extensively in other contexts [5,6], we can also relate curvature commutativity properties for these manifolds to other properties such as Einstein, self-dual, anti-self-dual and Osserman.…”

“…The notion of conformally Osserman is defined using the conformal Jacobi operator. One has that M is conformally Osserman ⇔ M is either self-dual or anti-self-dual [5]. If f = f (x 1 , x 2 , x 3 , x 4 ), let f /i := ∂ xi f and let f /ij := ∂ xi ∂ xj f .…”

Computation of partially invariant solutions for the Einstein Walker manifolds’ identifying equations

“…It is worthwhile to say that some other special cases have been considered in many references which yield some results about the structures admitted by these manifolds [14][15][16].…”