  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.
