Construction of Harmonic Maps Between Pseudo-Riemannian Spheres and Hyperbolic Spaces

Abstract

Defined here is an orthogonal multiplication for vector spaces with indefinite nondegenerate scalar product. This is then used, via the Hopf construction, to obtain harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces. Examples of harmonic maps are constructed using Clifford algebras. I. HARMONIC MAPS BETWEEN PSEUDo-RIEMANNIAN MANIFOLDS (1.1). In 1972 R. T. Smith ([S]) noticed that so-called orthogonal multiplications gave nice harmonic maps by applying the Hopf construction. This construction has not been done for vector spaces with an indefinite scalar product. There is a growing interest in physics in harmonic maps between pseudoRiemannian manifolds, especially since they have applications in string theory. For this reason it is useful to apply the Hopf construction to pseudo-Riemannian spheres and hyperbolic spaces to obtain new harmonic maps. In Parts I and II we shall give a theoretical background for the construction of harmonic maps. Many of the properties shown for pseudo-Riemannian manifolds are transcriptions of those from the Riemannian case in that we follow the results of [B]. For a review of the general properties of harmonic maps and the techniques used in this theory, see [ELI] and [EL2]. All the manifolds and maps considered in this paper are of the class C?? unless otherwise specified. 1.2. Let (M, g), (N, h) be pseudo-Riemannian manifolds and let 0: M N be a map from M to N; then one can construct a bundle of 1-forms on M with values in the pull-back bundle Q 'TN: T*M ? @+ (TN). This bundle is equipped with the connection V induced by the Levi-Civita connections on TM and TN. The covariant derivative of the differential Received by the editors April 17, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C50; Secondary 58E20.