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IntroductionLet M be a complex-n-dimensional indefinite Kfihler manifold, that means M is endowed with an almost complex structure J and with an indefinite Riemannian metric g, which is J-Hermitian, i.e., for all mEM g (Ju, Jv)=g(u,v) for all u,v~T"M, and VJ=O,where 17 denotes the Levi-Civita connection of g. It follows then, that J is integrable and the index of g is an even number 2s with 0 < s-< n. (The manifolds, maps .... will be understood to be of class CO~ Given these data, we shall use all the well-known conceigts for complex resp. for indefinite Riemannian manifolds. In particular the curvature tensor R of M satisfies :Ruv is a G-linear skew-adjoint operator of (T,,M, Or,), Ruv= -R~u and R~w+ Rvwu+ Rwuv=O (1.2) for all meM and u, v, w~T,,M. For brevity we will write u* for Ju, u^v for spanR(u, v) and g for g,.. A plane (this will allways mean a real-2-dimensional vector subspace) P of T,,M is nondegenerate (with respect to g) if and only if P has a basis {u, v} with A(u, v)= g(u, u)g(v, v)-g(u, 0 2 :~ 0 (1.3) (and we sometimes denote such a P by P+, PS, P+ if gIP • P is positive definite, negative definite or indefinite respectively). The sectional curvature function K is defined for a nondegenerate plane P of T,.M as usual by [-see (1.3)] K(P)=g(RuvV, U)/A(u,v), if P=uAv.(1.4)The restriction of K to the nondegenerate holomorphic resp. totally real planes is called the holomorphic resp. totally real sectional curvature (function) of M.A holomorphic plane is nondegenerate if and only if it contains some u with g(u, u)~ 0 and we write then Hu for K(u ^ u*).

Given a real hypersurface of a complex hyperbolic space CH', we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H~ "+ 1. Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of CH ".Geometriae Dedicata 20 (1986) 245-261.

We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrödinger equation showing the solitonic nature of those.

In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L3, with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L3. In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L3, i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L3. So, the term ‘regular’ means free of branch points. As in the minimal case, a conformal structure is naturally induced on M, which becomes a Riemann surface S. The corresponding isometric immersion is then conformal on S, and it does not have any singular points on S (i.e. points on which the differential of the mapping is not one-to-one). This is the way in which generalized maximal surfaces include regular ones. Moreover, branch points are the singular points of the conformal mapping on S. Whereas branch points of generalized minimal surfaces are isolated, we shall show in Section 2 that, in addition to isolated branch points, a generalized maximal surface in L3. may have non-isolated ones, in fact they constitute a 1-dimensional submanifold in a certain open subset of S (see Section 2). So our purpose is two-fold, firstly to study and explain in detail the branch points, and secondly to state several geometric results involving prescribed behaviour of those points on the surface.

Abstract. Semi-Riemannian manifolds with a suitable set of conformal symmetries are shown to be complete. Locally warped products are studied and warped-completeness is introduced. In the case of definite and complete basis, several assumptions on the growth of the warping function yield some of the three kinds of completeness. The case of 1-dimensional basis (including a known family of relativistic space-times) is specially studied. Null warped-completeness is related to the completeness of a certain conformal metric on the basis. Several examples and counter-examples explaining the main results are also given. (1991). 53C22, 53C50. Mathematics Subject Classifications

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We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.Comment: 20 pages, LaTe