Let $Gr(n, V)$ be the Grassmannian parametrising $n$-dimensional subspaces of a vector space $V$.
When $V$ is an inner product space, formulae exist for calculating the geodesic distance between points in $Gr(n, V)$ (e.g. Differential Geometry of Grassmannian Manifolds, Wong, 1967).
Now suppose $V$ is a equipped with a bilinear form of signature $(2, n)$ and let $Gr^+(2, V)$ be the Grassmannian parametrising positive definite planes in V.
Do similar formulae exist for $Gr^+(2, V)$? How about for distances to embedded Schubert varieties?
(Forgive me if this is completely obvious: I’m not a differential geometer.)
