| A typical path integral on a manifold, M is an informal expression of the form 1Zs∈ HMf se-Es Ds, where H(M) is a Hilbert manifold of paths with energy E( s ) < infinity, f is a real valued function on H(M), Ds is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret Ds as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, P of [0,1], let HP (M) be the finite dimensional Riemannian sub-manifold of H(M) consisting of piecewise geodesic paths adapted to P . Under certain curvature restrictions on M, it is shown that 1ZPe- 12Es dVolHPs →rs dns as mesh P→0, where ZP is a "normalization" constant, E : H( M) → [0,infinity) is the energy functional, VolHP is the Riemannian volume measure on HP (M), n is Wiener measure on continuous paths in M, and r is a certain density determined by the curvature tensor of M. |
