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*).

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Abstract. We present a theorem of Lancret for general helices in a 3-dimensional real-space-form which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3-sphere: the problem of solving natural equations and the closed curve problem.

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.

The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2, 1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is completely determined. Essentially, the solutions are warped products of orbits of the 1-dimensional groups of isometries and elastic curves in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The main tools are the equivalence of the two-dimensional O(2, 1) Nonlinear Sigma Model and the Willmore problem, and the description of the surfaces with rotational symmetry. A complete classification of such surfaces is obtained in this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a space-like axis is presented. The description of this new class of surfaces is based on a technique of surgery and a gluing process, which is illustrated by an algorithm. MSC 2000 Classification: Primary 53C40; Secondary 53C50

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

Let S be an immersed compact surface into a Riemannian manifold M. We denote by H and K the mean curvature vector field of S and the sectional curvature function of M with respect to the tangent space of S and defineformula here

In spaces with constant curvature, magnetic trajectories of a Killing magnetic field are showed to be centerlines of Kirchhoff elastic rods. If the space is also simply connected, then a Killing magnetic flow is proved to be equivalent with the Kirchhoff elastic rod variational model. This allows to see Killing magnetic flows as vortex filament flows and so use the Hasimoto transformation to state the solitonic nature of the magnetic trajectories as solutions of the cubic Schrödinger equation.