We present a simple, computation free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold which is invariant and normally hyperbolic persists under small perturbations of the diffeomorphism. The persistence of a Lipschitz invariant submanifold follows from an application of the Schauder fixed point theorem to a graph transform, wh...

The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in E4 with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in E4. Mathematics Subject Cl...

We prove that a germ of a holomorphic map f between Cn and Cn ′ sending one real-algebraic submanifold M ⊂ Cn into another M ′ ⊂ Cn ′ is algebraic provided M ′ contains no complex-analytic discs and M is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.

The submanifold quantum mechanics was opened by Jensen and Koppe (Ann. Phys. 63 (1971) 586-591) and has been studied for these three decades. This article gives its more algebraic definition and show what is the essential of the submanifold quantum mechanics from an algebraic viewpoint. MCS Codes: 34L40, 35Q40, 81T20, 32C25

The effective dynamic of a quantum particle on the tubular neighbourhood of some closed submanifold of a riemannian manifold can be described by an effective dynamic on the submanifold as the diameter of the tube decreases to zero. It depends whether Dirichletor Neumann conditions are imposed at the boundary of the tube.

We improve Chen-Ricci inequalities for a Lagrangian submanifold Mn of dimension n (n 2) in a 2n -dimensional complex space form M̃2n(4c) of constant holomorphic sectional curvature 4c with a semi-symmetric metric connection and a Legendrian submanifold Mn in a Sasakian space form M̃2n+1(c) of constant φ -sectional curvature c with a semi-symmetric metric connection, respectively.