A class of vector fields on path spaces

In this paper we show that the vector field X(del,h) on a based path space W-o(M) over a Riemannian manifold il I defined by parallel translating a curve h in the initial tangent space T(o)M via an affine connection del induces a solution flow which preserves the Wiener measure on the based path space W-o(M), provided the affine connection del is adjoint skew-symmetric. In the case when del is a metric connection, then del is adjoint skew-symmetric if and only if del is torsion skew-symmetric. (C) 1997 Academic press.