Total absolute horospherical curvature of submanifolds in hyperbolic space

We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of M, which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern-Lashof inequality for surfaces in 3-space and the horospherical Fenchel and Fary-Milnor`s theorems.

NEW CHARACTERIZATIONS OF COMPLETE SPACELIKE SUBMANIFOLDS IN SEMI-RIEMANNIAN SPACE FORMS

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau`s technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

Curvature Estimates for Submanifolds in Warped Products

We give estimates of the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.

Construction of Hamiltonian-Minimal Lagrangian submanifolds in Complex Euclidean Space

We describe several families of Lagrangian submanifolds in complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.

Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by the main result in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149-181], and with such a submanifold M and a point x in M we associate a canonical homogeneous structure I" (x) (a certain bilinear map defined on a subspace of T (x) M x T (x) M). We prove that I" (x) , together with the second fundamental form alpha (x) , encodes all the information about M, and we deduce from this the rigidity result that M is completely determined by alpha (x) and (Delta alpha) (x) , thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149-181] to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of I" (this result also seems to close a gap in [U. Christ, J. Differential Geom., 62 (2002), 1-15]). Here an important tool is the introduction of affine root systems of isoparametric submanifolds.

Submanifolds of dimension n-1 in [epsilon]n̳ with normals satisfying a Lipschitz condition.

"Paper written under Contract Nonr. 58304."

Warped Product Semi-Slant Submanifolds of a Sasakian Manifold

2000 Mathematics Subject Classification: 53C40, 53C25.

Geometry of Warped Product Semi-Invariant Submanifolds of a Locally Riemannian Product Manifold

2000 Mathematics Subject Classification: 53C42, 53C15.

Totally Umbilical Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

2000 Mathematics Subject Classification: 53C40, 53B25.

Warped Product CR-Submanifolds in Lorentzian para Sasakian Manifolds

2000 Mathematics Subject Classification: 53C15, 53C40, 53C42.

Contact CR-Submanifolds of Kenmotsu Manifolds

2000 Mathematics Subject Classification: 53C15, 53C42.

Éclatement et contraction lagrangiens et applications

Soit (M, ω) une variété symplectique. Nous construisons une version de l’éclatement et de la contraction symplectique, que nous définissons relative à une sous-variété lagrangienne L ⊂ M. En outre, si M admet une involution anti-symplectique ϕ, et que nous éclatons une configuration suffisament symmetrique des plongements de boules, nous démontrons qu’il existe aussi une involution anti-symplectique sur l’éclatement ~M. Nous dérivons ensuite une condition homologique pour les surfaces lagrangiennes réeles L = Fix(ϕ), qui détermine quand la topologie de L change losqu’on contracte une courbe exceptionnelle C dans M. Finalement, on utilise ces constructions afin d’étudier le packing relatif dans (ℂP²,ℝP²).