# American Institute of Mathematical Sciences

September  2017, 9(3): 291-316. doi: 10.3934/jgm.2017012

## Probability measures on infinite-dimensional Stiefel manifolds

 1 Microsoft Deutschland GmbH, Germany 2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italia

* Corresponding author: A. C. G. Mennucci

‡ Eleonora Bardelli contributed to this paper in her personal capacity. The views expressed in this paper are those of the authors and do not necessarily reflect the views of Microsoft Corporation

Received  December 2015 Revised  June 2016 Published  June 2017

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 $S\subset H$ be the sphere, $p\in S$. Let $\mu$ be the push forward of a Gaussian measure $\gamma$ from $T_p S$ onto $S$ using the exponential map. Let $v\in T_p S$ 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 $M\subset H$ 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.

Citation: Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012
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##### References:
Proof of Proposition 3.6. Two points on the ellipsoid $C$, their images under $\exp_{-p}$ and, in white, their images under $\exp_p$. The black diamonds on the sphere can coincide with the white diamonds only if they all lie on the equator.
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