Classical invariants of Legendrian knots in the 3-dimensional torus

All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $xi$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight oriented contact structure on $T^3$ using projections.

Locally convex hypersurfaces immersed in $H^n times R$

We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to $R^n$. In the latter case it is either a vertical graph over a convex domain in $H^n$ or has what we call a simple end.

Manifolds that are not leaves of codimension one foliations

We present new open manifolds that are not homeomorphic to leaves of any C^0 codimension one foliation of a compact manifold. Among them are simply connected manifolds of dimension 5 or greater that are non-periodic in homotopy or homology, namely in their 2-dimensional homotopy or homology groups.

Normal subgroups of diffeomorphism and homeomorphism groups of R^n and other open manifolds

We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

Riemannian manifolds not quasi-isometric to leaves in codimension one foliations

Every open manifold L of dimension greater than one has complete Riemannian metrics g with bounded geometry such that (L,g) is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the local geometry of (L,g) suffice to make it quasi-isometric to a leaf of such a foliation. We introduce the `bounded homology property, a semi-local property of (L,g) that is necessary for it to be a leaf in a compact manifold in codimension one, up to quasi-isometry. An essential step involves a partial generalization of the Novikov closed leaf theorem to higher dimensions.