Harmonic 4-spaces

Abstract

[9]. Repeated indices represent contractions with the metric g, ~ denotes symmetrization over all free indices, and k a is a constant. It has been conjectured that a harmonic space with positive definite metric is flat or locally rank one symmetric. In dimensions 2 and 3, this follows from only H 1 ~ it is also valid for a 4-manifold, but the proofs to date [2; 9] use H 1, H2, and H 3 and rely heavily on the constancy of k a. The present work provides an alternative approach to the problem in 4 dimensions by using group representations to study the pointwise significance of the H a. Consequent ly , our results make no assumpt ion that the scalar func t ions k a be constant . Now H1 is simply the Einstein condition, so restricting to a pseudoRiemannian Einstein 4-manifold, we consider separately H 2 and H 3. The idea of searching for manifolds satisfying just some of the H a has its origin in [4]. Having first discussed curvature, we are able to relate H 2 to the notion of self-duality. With the aid of some classical invariant theory, we also show that H 1 and H 3 are sufficient to conclude that M is flat or locally rank one symmetric, except when the metric has signature 0.