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*).

No abstract

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