  We give time-slicing path integral formulas for solutions to the heat
equation corresponding to a self-adjoint Laplace type operator acting on
sections of a vector bundle over a compact Riemannian manifold with boundary.
More specifically, we show that such a solution can be approximated by
integrals over finite-dimensional path spaces of piecewise geodesics
subordinated to increasingly fine partitions of the time interval. We consider
a subclass of mixed boundary conditions which includes standard Dirichlet and
Neumann boundary conditions.
