We study deformations of associative submanifolds Y 3 ⊂ M of a G2 manifold M . We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative submanifolds of M . The local equations at each associative Y are restrictions of a global equat...

By analysing supersymmetry transformations we derive new BPS equations for the D=11 fivebrane propagating in flat space that involve the world-volume three-form. The equations generalise those of 2,3,4 and 5 dimensional special Lagrangian submanifolds and are relevant for describing membranes ending on these submanifolds.

We investigate biharmonic submanifolds of the product of two space forms. We prove a necessary and sufficient condition for biharmonic submanifolds in these product spaces. Then, we obtain mean curvature estimates for proper-biharmonic submanifold of a product of two unit spheres. We also prove a non-existence result in the case of the product of a sphere and a hyperbolic space.

A family of compact submanifolds with boundary has a nonempty stable interior intersection in R provided a certain geometric estimate holds. As a global application we establish a sharp intersection criterion for a family of properly embedded mj -dimensional submanifolds Mj ⊂ R that is based solely on their Gauss maps Gj : Mj → G(n−mj , n) , Gj(p) = [TpMj ]⊥ .

We study half lightlike submanifold M of an indefinite transSasakian manifold such that its structure vector field is tangent to M . First we study the general theory for such half lightlike submanifolds. Next we prove some characterization theorems for half lightlike submanifolds of an indefinite generalized Sasakian space form.

In this paper, we study half-lightlike submanifolds of a semi-Riemannian manifold such that the shape operator of screen distribution is conformal to the shape operator of screen transversal distribution. We mainly obtain some results concerning the induced Ricci curvature tensor and the null sectional curvature of screen transversal conformal half-lightlike submanifolds. 2010 Mathematics Subje...

Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of N × R. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form.

In this paper, we study the curvature of a semi-Riemannian manifold M̃ of quasi-constant curvature admits some half lightlike submanifolds M . The main result is two characterization theorems for M̃ admits extended screen homothetic and statical half lightlike submanifolds M such that the curvature vector field of M̃ is tangent to M .

By making use of the symplectic reduction and the cohomogeneity method, we give a general method for constructing Hamiltonian minimal submanifolds in Kaehler manifolds with symmetries. As applications, we construct infinitely many nontrivial complete Hamiltonian minimal submanifolds in CP n and Cn.

Cosymplectic manifolds provide a natural setting for time dependent mechanical systems as they are locally product of a Kaehler manifold and a one dimensional manifold. Thus study of warped product submanifolds of cosymplectic manifolds is significant. In this paper we have proved results on the non-existence of warped product submanifolds of certain types in cosymplectic manifolds. M.S.C. 2000...