  Let M be a smooth closed (compact without boundary) Riemannian manifold of
dimension n and P a q-dimensional smooth submanifold of M. U will denote the
tubular neighborhood of P in M. Let E be a smooth vector bundle over M. Here we
will obtain a vector bundle Generalized Feynman-Kac formula associated to U and
the vector bundle differential operator L consisteing of half the generalized
Laplacian, a vector field X on M and a potential term V on M. From this
formula, we shall deduce the usual Feynman-Kac formula as well as a stochastic
representation of the Generalized Elworthy-Truman Heat Kernel formula, and
ultimately the heat kernel formula. The Feynman-Kac expression can be expanded
and from this expansion we shall deduce both the generalized heat trace and
heat content expansions. The Generalized Feynman-Kac Formula is thus at the
center of several other previously known results.
