We show that the moduli spaces of associative submanifolds of a G 2 manifold (and Cayley submanifolds of a Spin(7) manifold) can be perturbed to smooth manifolds. By using connections as natural parameters and by constraining them with an additional equation and using Seiberg-Witten theory we can make them compact, and hence assign local invariants to these submanifolds. The local equations of ...

Abstract A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization the so-called isoparametric submanifolds. In this paper, we prove regular fibers an analytic map π : M m+k → B k whenever endowed with complete Finsler metric and there restriction submersion for certain on image. addition, when provide si...

A calibration ¢ is a differential form on a Riemannian manifold with two additional properties. First, the form should be closed under exterior differentiation. Second, it should be less than or equal to the volume form on each oriented submanifold (of the same dimension as the degree of the form ¢). Each calibration ¢ determines a geometry of submanifolds, namely those oriented submanifolds fo...

A calibration ¢ is a differential form on a Riemannian manifold with two additional properties. First, the form should be closed under exterior differentiation. Second, it should be less than or equal to the volume form on each oriented submanifold (of the same dimension as the degree of the form ¢). Each calibration ¢ determines a geometry of submanifolds, namely those oriented submanifolds fo...

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces bear natural structures ...

Our paper aims to study the geometry of submanifolds an almost-complex metallic manifold. We give example this type manifold, and reveal fundamental properties structure induced on submanifolds. establish subsets in manifolds, such as invariant, anti-invariant, slant

B.Y. Chen [3] established a sharp inequality for the warping function of a warped product submanifold in a Riemannian space form in terms of the squared mean curvature. For a survey on warped product submanifolds we refer to [4]. In [8], we established a similar relationship between the warping function f (intrinsic structure) and the squared mean curvature and the holomorphic sectional curvatu...

We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in CPn via the Hopf fibration; and get new examples of biharmonic submanifolds in S as byproducts .