We consider the dynamic property of volume preserving mean curvature flow. This flow was introduced by Huisken (J Reine Angew Math 382:35–48, 1987) who also proved it converges to a round sphere same enclosed if initial hypersurface is strictly convex in Euclidean space. study stability this hyperbolic In particular, we prove that hyperbolically and close an umbilical $$L^2$$ -sense, then exist...

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 ps...

We first prove some results on invariant lightlike submanifolds of indefinite Sasakian manifolds. Then, we introduce a general notion of contact Cauchy-Riemann (CR) lightlike submanifolds and study the geometry of leaves of their distributions. We also study a class, namely, contact screen Cauchy-Riemann (SCR) lightlike submanifolds which include invariant and screen real subcases. Finally, we ...

In this paper we prove that there do not exist warped product GCR-lightlike submanifolds in the form M = N⊥ ×λ NT such that N⊥ is an anti-invariant submanifold tangent to V and NT an invariant submanifold of M̄ , other than GCR-lightlike product in an indefinite cosymplectic manifold. We also obtain some characterizations for a GCR-lightlike submanifold to be locally a GCR-lightlike warped produ...

Copyright q 2012 Varun Jain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We define GCR-lightlike submanifolds of indefinite cosymplectic manifolds and give an example. Then, we study mixed geodesic GCR-lightlike submanif...

‎based on a definition for circle in finsler space‎, ‎recently proposed by one of the present authors and z‎. ‎shen‎, ‎a natural definition of extrinsic sphere in finsler geometry is given and it is shown that a connected submanifold of a finsler manifold is totally umbilical and has non-zero parallel mean curvature vector field‎, ‎if and only if its circles coincide with circles of the ambient...

In this paper, we study the geometry of half lightlike submanifolds of an indefinite Sasakian manifold. There are several different types of half lightlike submanifolds of an indefinite Sasakian manifold according to the form of its structure vector field. We study two types of them here: tangential and ascreen half lightlike submanifolds.

introduction with the growing popularity of totally extraperitoneal (tep) laparoscopic inguinal hernia repair, many surgeons will inevitably face the clinical conundrum of how to simultaneously repair an umbilical hernia and an inguinal hernia via tep approach. case presentation in this case report, we describe the successful use of the umbilical hernia defect to insert the balloon dissector an...