For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, $\Psi$-submanifold is defined as submanifold $N$ $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The class $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups compact groups. carries natural algebra structure, given by Frolich...

where Xo is a constant vector and AX,-t — XitX{t, t = 1, 2, . . . , k. If in particular all eigenvalues {Atl, . . . , A,t} are mutually different, then M is said to be of i-type. A Jfc-type submanifold is said to be null if one of the A;t, t — 1, 2, . . . , k, is null. It is easy to see that if M is compact, then Xo in (1.1) is exactly the centre of mass in E . A submanifold M of a hypersphere ...

In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together noncommutative embeddings. We show that basic concepts, such as the second fundamental form and Weingarten map, translate into setting and, in particular, prove a analogue Gauss’ equations for curvature submanifold. Moreover, mean an embedding is readily introduced, giving natural definition min...

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which characterized by the condition that second form proportional to metric. It also generalization of notion geodesic submanifold. In this paper, we classify congruent classes full submanifolds non-flat space forms, and consider their moduli spaces. As consequence, show some spaces isometric immersions bet...

For this quarter of century, quantum differential operators in a lower dimensional submanifold embedded or immersed in real n-dimensional euclidean space E n have been studied as physical models, which are realized as restriction of the operators in E n to the submanifold. For this decade, the Dirac operators in the submanifold have been investigated, which are identified with operators of the ...