Contents 1. Introduction 2 Part 1. Classical and almost classical results 2 1.1. The Gauss map 3 1.2. Minimal graphs 3 1.3. The maximum principle 5 2. Monotonicity and the mean value inequality 6 3. Rado's theorem 8 4. The theorems of Bernstein and Bers 9 5. Simons inequality 10 6. Heinz's curvature estimate for graphs 10 7. Embedded minimal disks with area bounds 11 8. Stable minimal surfaces ...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone Lagrangian submanifold vanishes in the homology of the ambient symplectic manifold. Combining this with spectral invariants we provide a new mechanism for proving Lag...

In this paper, we study submanifolds in a Riemannian manifold with a semi-symmetric non-metric connection. We prove that the induced connection on a submanifold is also semi-symmetric non-metric connection. We consider the total geodesicness and minimality of a submanifold with respect to the semi-symmetric non-metric connection. We obtain the Gauss, Cadazzi, and Ricci equations for submanifold...

In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2. To appear in the Proceedings volume for the conference V...

Certain geometric properties of submanifolds of configuration space are numerically investigated for classical phi(4) models in one and two dimensions. Peculiar behaviors of the computed geometric quantities are found only in the two-dimensional case, when a phase transition is present. The observed phenomenology strongly supports, though in an indirect way, a recently proposed topological conj...

We deal with a CR-submanifold M of a para-Kählerian manifold M, which carries a J-skew-symmetric vector field X. It is shown that X defines a global Hamiltonian of the symplectic form Ω on M and JX is a relative infinitesimal automorphism of Ω. Other geometric properties are given. 1. Introduction. CR-submanifolds M of some pseudo-Riemannian manifolds M have been first investigated by Rosca [10...

Questions of isolation phenomena for minimal submanifolds have been posed for many years. Perhaps the most studied case is for minimal submanifolds of the sphere. Lawson [L1], Chern, do Carmo and Kobayashi [CCK], Barbosa [B], Fischer-Colbrie [FC] and others studied minimal submanifolds of the sphere using a range of techniques and obtained existence and uniqueness results. An important part of ...

Pseudo-parallel (shortly PP) submanifolds are defined as a generalization of semi-parallel (shortly SP ) submanifolds and as extrinsic analogue of pseudo-symmetric (shortly PS) manifolds (in the sense of R. Deszcz) [1], [26]. Asperti et al [1] obtained a description of PP hypersurfaces in space form as quasiumbilic hypersurfaces or cyclids of Dupin and studied PP surfaces with maximal first nor...

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regularity theorem of Ecker and Huisken for Lipschitz hypersurfaces. In particular, any ...