Rabah Souam

The Schläfli formula for polyhedra and piecewise smooth hypersurfaces

Abstract The classical Schlafli formula relates the variations of the dihedral angles of a smooth family of polyhedra in a space form to the variation of the enclosed volume. We extend here this formula to immersed piecewise smooth hypersurfaces in Einstein manifolds. This leads us to introduce a natural notion of total mean curvature of piecewise smooth hypersurfaces and a consequence of our formula is, for instance, in Ricci-flat manifolds, the invariance of the total mean curvature under bendings. We also give a simple and unified proof of the Schlafli formula for polyhedra in Riemannian and pseudo-Riemannian space forms. Moreover, we show that the formula makes sense even for polyhedra which are not necessarily embedded.

Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient condition under which a Riemannian three-manifold carrying a unit Killing field admits totally geodesic surfaces and we study local and global properties of three-manifolds satisfying this condition.

THE CLASSIFICATION OF TOTALLY UMBILICAL SURFACES IN HOMOGENEOUS 3-MANIFOLDS

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.