Exercise 11, § 4

Let H be the subset of $\mathbf{R}^n$ consisting of the points $x = (x_0, \ldots, x_{n-1})$ for which the form

is < 0, and let $\mathbf{P}H$ be the image of H in the projective space $\mathbf{P}_{n-1}(\mathbf{R})$. Let G be the orthogonal group of the form B.

a) Show that $\mathbf{P}H$ is a homogeneous space of G, that the stabiliser of a point is compact, and that G acts properly on $\mathbf{P}H$.

b) Let $\omega$ and $\Omega$ be the differential forms on H defined by the formulas

Show that $\Omega$ is the inverse image under the canonical projection $\pi : H \to \mathbf{P}H$ of a differential form $\tilde{\Omega}$ on $\mathbf{P}H$. Show that the positive measure $\nu$ associated to $\tilde{\Omega}$ (Differentiable Varieties R, 2nd part) is invariant under G, and that it is the only such measure, up to a scalar factor.

c) Let C be an open simplicial cone with vertex 0 in $\mathbf{R}^n$ (§ 1, no. 6). Assume that C is contained in H, and denote by $\mathbf{P}C$ the image of C under $\pi : H \to \mathbf{P}H$. Show that, if $n \geq 3$, then $\nu(\mathbf{P}C) < \infty$ (identify $\mathbf{P}H$ with the subspace of $\mathbf{R}^{n-1}$ consisting of the $(x_1, \ldots, x_{n-1})$ such that $\sum_{i=1}^{n-1} x_i^2 < 1$, and determine the measure corresponding to $\nu$ on this subspace). Show that $\mathbf{P}C$ is relatively compact in $\mathbf{P}H$ if and only if $\overline{C}$ is contained in H.