Complex IP pseudo-Riemannian algebraic curvature tensors

Abstract

1. Introduction. The Riemann curvature tensor contains a great deal of information about the geometry of the underlying pseudo-Riemannian manifold; pseudo-Riemannian geometry is to a large extent the study of this tensor and its covariant derivatives. It is often convenient to work in a purely algebraic setting. We shall say that a tensor is an algebraic curvature tensor if it satisfies the symmetries of the Riemann curvature tensor. The Riemann curvature tensor defines an algebraic curvature tensor at each point of the manifold; conversely every algebraic curvature tensor is locally geometrically realizable. Thus algebraic curvature tensors are an integral part of certain questions in differential geometry.The skew-symmetric curvature operator is a natural object of study; there are other natural operators and we refer to [4] for a survey of this area. In the present paper, we examine when the (complex) Jordan normal form of the skew-symmetric curvature operator is constant either in the real or complex settings. In Section 2, we give the basic definitions and notational conventions we shall need. We also review the basic results in the real setting. In Section 3, we discuss a natural generalization to the complex setting.

Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

“…The IP algebraic curvature tensors were also extensively studied in pseudoriemannian and in complex settings. We refer to [8,9,10,11] for results in these directions.…”