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This paper gives a rigorous interpretation of a Feynman path integral on aRiemannian manifold M with non-positive sectional curvature. A $L^2$ Riemannianmetric $G_P$ is given on the space of piecewise geodesic paths $H_P(M)$ adaptedto the partition $P$ of $[0,1]$, whence a finite-dimensional approximation ofWiener measure is developed. It is proved that, as $mesh(P) \to 0$, theapproximate Wiener measure converges in a $L^1$ sense to the measure$e^{-\frac{2 + \sqrt{3}}{20\sqrt{3}} \int_0^1 Scal(\sigma(s)) ds} d\nu(\sigma)$on the Wiener space $W(M)$ with Wiener measure $\nu$. This gives a possibleprescription for the path integral representation of the quantized Hamiltonian,as well as yielding such a result for the natural geometric approximationschemes originating in [L. A. Andersson and B. K. Driver, J. Funct. Anal. 165(1999), no. 2, 430-498] and followed by [Adrian P. C. Lim, Rev. Math. Phys. 19(2007), no. 9, 967-1044].

An Approximation to Wiener Measure and Quantization of the Hamiltonian on Manifolds with Non-positive Sectional Curvature