A Class of Vector Fields on Path Spaces

In this paper we show that the vector fieldX∇, hon a based path spaceWo(M) over a Riemannian manifoldMdefined by parallel translating a curvehin the initial tangent spaceToMvia an affine connection ∇ induces a solution flow which preserves the Wiener measure on the based path spaceWo(M), provided the affine connection ∇ is adjoint skew-symmetric. In the case when ∇ is a metric connection, then ∇ is adjoint skew-symmetric if and only if ∇ is torsion skew-symmetric.