Harmonic 4-spaces (Q1056914)

In this article, the author gives a representation theoretical proof of the locally symmetric character of 4-dimensional harmonic spaces. This approach allows him to treat the case of pseudo-Riemannian manifolds in a unified way. The article contains several useful lemmata of general interest on the decomposition into irreducible subspaces of the spaces of curvature tensors and their covariant derivatives, and discusses how they relate to geometric assumptions. The proof involves applying classical invariant theory to pointwise algebraic conditions implied by the harmonicity of the space.