  In this paper, we consider optimization problems w.r.t. to pairs of
orthogonal matrices $XY = I$. Problems of this form arise in several
applications such as finding shape correspondence in computer graphics. We show
that the space of such matrices is a Riemannian manifold, which we call the
biorthogonal manifold. To our knowledge, this manifold has not been studied
before. We give expressions of tangent space projection, exponential map, and
retraction operators of the biorthogonal manifold, and discuss their numerical
implementation.
