Construction of harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces

Abstract

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-Riemanñian 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°°u nless otherwise specified. This bundle is equipped with the connection V induced by the Levi-Civita connections on TM and TN. The covariant derivative of the differential

Classification of pseudo-Riemannian submersions with totally geodesic fibres from pseudo-hyperbolic spaces

“…We now show that the case r = 7 is not possible, namely, we see that there is no pseudo-Riemannian submersion π : H 31 15 → H 16 8 (−4) with connected totally geodesic fibres. By Ranjan [39], the linear map U :…”

“…A Hopf construction is a map ϕ : R p × R q → R n+1 defined by ϕ(x, y) = (|x| 2 − |y| 2 , 2G(x, y)) for some orthogonal multiplication G (see [5,41]). The Hopf construction can provide several examples of harmonic morphisms (see [31,41]), and we would like to refer the reader to the beautiful book [5] due to Baird and Wood for other nice results on this topic. Since the sectional curvatures K, K of the total and base spaces of any pseudo-Riemannian submersion between real space forms must obey K = 4K, we are forced to consider the map ϕ(x, y)/2 instead.…”

“…d−1 will give the same π 7 , π 8 , π 9 . In [31], Konderak constructed the harmonic morphisms 2π 7 and 2π 8 via the Hopf construction (see also [5,). For identification (3.11) 16 , the Hopf pseudo-Riemannian submersion π 9 : H 15 7 → H 8 4 (−4) can be written explicitly as…”

Spheres and circles in the tangent space at a point on a Riemannian manifold with respect to an indefinite metric

“…The models of a sphere and the relations between spheres and hyperboloids, between spheres and cones are of certain interest in pseudo-Riemannian geometry. Another current problem is the obtaining of the correspondence between a circle and other quadratic curves (for instance [1,7,8,9,11,12]).…”

Non-compact Hopf maps and fuzzy ultra-hyperboloids

“…2 See also the preceding literatures [47,48] for the 1st and 2nd split Hopf maps. 3 The hybrid Hopf maps are a hybridization of the compact and split Hopf maps in the sense that the total manifolds are same as those of the split Hopf maps and the fibres are those of the compact Hopf maps.…”