A submanifold of R n whose tangent space makes constant angle with a fixed direction d is called a helix. In the first part of the paper we study helix hypersurfaces. We give a local description of how these hypersurfaces are constructed. As an application we construct (non flat) minimal helices hypersurfaces in R n for n > 3 . In the second part we give a characterization of helix submanifolds related to the solutions of the so called eikonal differential equation. As a corollary we give necessary and sufficient conditions for a manifold M to be immersed as an helix in some Euclidean space. In the third part of this paper we study r -helices submanifolds. That is to say submanifolds such that its tangent space makes a constant angle with r linearly independent directions.

Let M \subset {\complex}^n be\ud a complex n-dimensional Hermitian symmetric space endowed with\ud the hyperbolic form \omega_{hyp}. Denote by (M^*, \omega_{FS}) the compact dual of (M, \omega_{hyp}), where\omega_{FS} is the Fubini--Study form on M^*. Our first result\ud is Theorem 1 where, with the aid of the theory of Jordan triple systems, we construct an explicit {\em symplectic\ud duality}, namely a diffeomorphism \Psi_M: M\rightarrow\ud {\real}^{2n}={\complex}^n\subset M^* satisfying\ud \Psi_M^*\omega_0=\omega_{hyp} and\ud \Psi_M^*\omega_{FS}=\omega_0 for the pull-back of \Psi_M, where \omega_0\ud is the restriction to M of the flat Kaehler form of\ud the Hermitian\ud positive Jordan triple system associated to M.\ud Amongst other properties of the\ud map \Psi_M, we also show that it takes (complete) complex and\ud totally geodesic submanifolds of $M$ through the origin to complex\ud linear subspaces of {\complex}^n. As a byproduct of the proof of\ud Theorem \ref{mainteor} we get an interesting characterization\ud of the Bergman form of a Hermitian\ud symmetric space in terms of its restriction to classical complex\ud and totally geodesic submanifolds passing through the origin

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We prove, in a purely geometric way, that there are no connected irreducible proper subgroups of SO(N, 1). Moreover, a weakly irreducible subgroup of SO(N, 1) must either act transitively on the hyperbolic space or on a horosphere. This has obvious consequences for Lorentzian holonomy and in particular explains clasification results of Marcel Berger's list (e.g. the fact that an irreducible Lorentzian locally symmetric space has constant curvatures). We also prove that a minimal homogeneous submanifold of hyperbolic space must be totally-geodesic.

Abstract. In this paper we study the homogeneous Kähler manifolds (h.K.m.) which can be Kähler immersed into finite or infinite dimensional complex space forms. On the one hand we completely classify the h.K.m. which can be Kähler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kähler immersions into the finite dimensional complex projective space to the infinite dimensional setting.