The properties of normal injectivity radius i(K, M) (thickness), of C submanifolds K of complete Riemannian manifolds M are studied. We introduce the notion of geometric focal distance for C submanifolds by using metric balls. A formula for i(K, M) in terms of the double critical points and the geometric focal distance is proved. The thickness of knots and ideal knots relate to the study of DNA...

An immersed submanifold f :M → R , n ≥ 3, into Euclidean space with the induced metric is called of rank two if at any point the kernel of its vector valued second fundamental form has codimension two. Equivalently, we have that the image of the Gauss map in the Grassmannian of non-oriented n-planes Gn is a surface. These submanifolds have been the object of a great deal of work in Riemannian G...

In this paper, we introduce some inequalities for screen homothetic half lightlike submanifolds of a real space form constant sectional curvature , endowed with semi-symmetric metric connection. Using these inequalities, derive characterizations such submanifolds. Finally, Chen-Ricci are calculated. Morever, the equality cases considered and get results.

In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist ...

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic RP 2m in CP n. When it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CP n (but not among all submanifolds in this Z/2 homology class). Also the totally geodesic RP 2m−1 minimizes volume in its Hamiltonian deformation class in CP n. As a...

In this paper we use Lie group actions on noncompact Riemannian manifolds with calibrations to construct calibrated submanifolds. In particular, if we have an (n−1)torus acting on a noncompact Calabi-Yau n-fold with a trivial first cohomology, then we have a special Lagrangian fibration on that n-fold. We produce several families of examples for this construction and give some applications to s...

This article grew out of several talks that the author presented at the Banach Institute and at the University of Bialystok in Poland during November of 2001. It describes six problems from the geometry of submanifolds. Some of the problems come from the theory of constant curvature submanifolds in Euclidean space, as well as applications of Morse theory of the height function to the problem of...

We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of s-representations.

We will give a construction to obtain canonically an “isotropic average“ of given C close isotropic submanifolds of a Kähler manifold. The isotropic average will be C close to the given submanifolds. The construction relies on an averaging procedure by Weinstein and on “Moser’s trick”.