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 subset of H be the sphere, p is an element of S. Let mu be the push forward of a Gaussian measure gamma from TpS onto S using the exponential map. Let v 2 TpS be a Cameron-Martin vector for gamma; let R be a rotation of S in the direction v, and nu = R# mu be the rotated measure. Then mu, nu are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron-Martin direction produces equivalent measures. Let gamma be a Gaussian measure on H; then there exists a smooth closed manifold M subset of H such that the projection of H to the nearest point on M is not well defined for points in a set of positive gamma measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability.