  Certain natural geometric approximation schemes are developed for Wiener
measure on a compact Riemannian manifold. These approximations closely mimic
the informal path integral formulas used in the physics literature for
representing the heat semi-group on Riemannian manifolds. The path space is
approximated by finite dimensional manifolds consisting of piecewise geodesic
paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of
piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian
structures $G^i_P$. It is proved that as the mesh of the partition tends to
$0$,
  $$ 1/Z_P^i e^{- 1/2 E(\sigma)} Vol_{G^i_P}(\sigma) \to
\rho_i(\sigma)\nu(\sigma) $$
  where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, and
for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is
the Riemannian volume form relative $G^i_P$, and $\nu$ is Wiener measure on
paths on $M$. Here $\rho_1 = 1$ and
  $$ \rho_0 (\sigma) = \exp( -1/6 \int_0^1 Scal(\sigma(s))ds ) $$
  where $Scal$ is the scalar curvature of $M$. These results are also shown to
imply the well know integration by parts formula for the Wiener measure.
