Let $V$ be a real vector bundle with non-degenerate metric whose index is $(p,q)$, i.e. $p$ ($q$) is the dimension of maximal subspace of positive(negative) definite. Let $G^+_p(V)$ be the all positive definite $p$-dim subspaces, and $G^+_p(x^\perp)$ be all positive definite $p$-dim subspaces lie in $x^\perp$ for some $x\in V$ with $x^2<0$.
Q: How to compute the dimension of $G^+_p(V)$ and codimension of $G^+_p(x^\perp)$ in $G^+_p(V)$ ?