A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we ...

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in C are established in this paper. The finite determination result gives sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for realanalytic generic submanifolds under appropriate assumptio...

A classical problem in the theory of minimal submanifolds of Euclidean spaces is to study the existence of a minimal submanifold with a prescribed behavior at infinity, or to determine from the asymptotes the geometry of the whole submanifold. Beyond the intrinsic interest of these questions, they are also of crucial importance when studying the possible singularities of minimal submanifolds in...

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel of submanifolds. We prove that the submanifolds are anisotropic-minimal obtain a general Cartan-type formula form with vanishing reversible torsion, from which give some classifications on number distinct principal curvatures or their multiplicities.

Biconservative submanifolds, with important role in mathematical physics and differential geometry, arise as the conservative stress-energy tensor associated to variational problem of biharmonic submanifolds. Many examples biconservative hypersurfaces have constant mean curvature. A famous conjecture Bang-Yen Chen on Euclidean spaces says that everybiharmonic submanifold has null Inspired by co...

By a theorem of Mclean, the deformation space of an associative sub-manifolds of an integrable G2 manifold (M, ϕ) at Y ⊂ M can be identified with the kernel of the Dirac operator D / : Ω 0 (ν) → Ω 0 (ν) on the normal bundle ν of Y. We generalize this to non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to mo...

We study deformations of associative submanifolds Y 3 ⊂ M 7 of a G 2 manifold M 7. 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 eq...

We introduce a very natural class of potential submanifolds in pseudo-Euclidean spaces (each Ndimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudoEuclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanifold. We show that all potential submanifolds bear natural special ...

In this article, pseudoparallel submanifolds for almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space are investigated. The is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as pseudoparallel, $2-$pseudoparallel, Ricci generalized and $2-$Ricci has been characterized. Necessary sufficient conditions given invariant to be total geodesic acco...