The relations among two transversal submanifolds and global manifold

TL;DR

This paper explores the geometric relations among two transversal submanifolds and a global manifold in Riemann geometry, linking their metric and curvature tensors, and discusses implications for matter fields and theories like Kaluza-Klein and string theory.

Contribution

It introduces a novel approach by replacing normal vectors with tangent vectors to relate submanifold geometries and extends these ideas to non-direct product manifolds in theoretical physics.

Findings

Relations among metric, Christoffel symbols, and curvature tensors are established.

Inner product of tangent vectors influences matter field energy-momentum tensors.

Results align with Kaluza-Klein theory and generalize models in string and D-brane theories.

Abstract

In Riemann geometry, the relations among two transversal submanifolds and global manifold are discussed. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols and curvature tensors of the three manifolds are linked together. When the inner product of the two tangent vectors vanishes, some corollaries of these relations give the most important second fundamental form and Gauss-Codazzi equation in the conventional submanifold theory. As a special case, the global manifold is Euclidean is considered. It is pointed out that, in order to obtain the nonzero energy-momentum tensor of matter field in a submanifold, there must be the contributions of the above inner product and the other submanifold. In general speaking, a submanifold is closely related to the matter fields of the other submanifold through the above inner product. This conclusion is in agreement with the Kaluza-Klein theory and it can be applied to generalize the models of direct product of manifolds in string and D-brane theories to the more general cases --- nondirect product manifolds.