  The space of all non degenerate bilinear structures on a manifold $M$ carries
a one parameter family of pseudo Riemannian metrics. We determine the geodesic
equation, covariant derivative, curvature, and we solve the geodesic equation
explicitly. Each space of pseudo Riemannian metrics with fixed signature is a
geodesically closed submanifold. The space of non degenerate 2-forms is also a
geodesically closed submanifold. Then we show that, if we fix a distribution on
$M$, the space of all Riemannia metrics splits as the product of three spaces
which are everywhere mutually orthogonal, for the usual metric. We investigate
this situation in detail.
