RELATIONS OF TWO TRANSVERSAL SUBMANIFOLDS AND GLOBAL MANIFOLD

Abstract

In Riemann geometry, the relations of two transversal submanifolds and global manifold are discussed without any concrete models. 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 connected at the intersection points of the two submanifolds. When the inner product of the two tangent vectors of submanifolds 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 which 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. Generally speaking, a submanifold is closely related to the matter fields of the other submanifold and the two submanifolds affect each other through the above inner product.