We consider functions with values in the algebra of Lorentz numbers L which are differentiable with respect to the algebraic structure of L as an analogue of holomorphic functions. Then we apply these functions to prove a Weierstrass representation theorem for Lorentz surfaces immersed in the space R 3 1 . In the proof we essentially follow the model of the complex numbers. We apply our representation theorem to construct explicit minimal immersions.

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

Abstract. We consider transversally harmonic foliated maps between two Riemannian manifolds equipped with Riemannian foliations. We give various characterisations of such maps and we study the relation between the properties "harmonic" and "transversally harmonic" for a given map. We also consider these problems for particular classes of manifolds: manifolds with transversally almost Hermitian foliations and Riemannian flows.2000 Mathematics Subject Classification. 53C12, 58E20.

We consider leaf preserving maps between manifolds equipped with Riemannian foliations. We construct a transversal tension field for such maps and define transversally harmonic maps. Then we give some examples of such maps using the suspension construction.

We consider a Riemannian manifold with a compatible f -structure which admits a parallelizable kernel. With some additional integrability conditions it is called (almost) K, C, S-manifold and is a natural generalization of the (almost) contact metric and the Sasakian manifolds. There are presented various methods of constructing examples of such manifolds. There are used structures on the principal bundles and the pull-back bundles. Then there are considered relations between (almost) K, C, S-manifolds and transverse almost Hermitian structures on the foliated manifolds. (2000): 53D15, 53C15, 57R30. Mathematics Subject Classification