Probability measures on infinite-dimensional Stiefel manifolds

An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that H is an infinite-dimensional separable Hilbert space. Let S⊂H be the sphere, p∈S. Let μ be the push forward of a Gaussian measure γ from TpS onto S using the exponential map. Let v∈TpS be a Cameron--Martin vector for γ; let R be a rotation of S in the direction v, and ν=R#μ be the rotated measure. Then μ,ν are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron--Martin direction produces equivalent measures. Let γ be a Gaussian measure on H; then there exists a smooth closed manifold M⊂H such that the projection of H to the nearest point on M is not well defined for points in a set of positive γ measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability.