Quotient of $GL(\mathbb{R}^n)$ by $O(\mathbb{R}^n)$

Question

I suppose the quotient $GL(\mathbb{R}^n)/O(\mathbb{R}^n)$ has manifold structure. Is there a name for this manifold? Google isn't helping find it.

Answer 1

You are almost surely familiar with the case $n=2$, where the quotient space $GL(\mathbb R^2) / O(\mathbb R^2)$ is the hyperbolic plane.

In general this example fits into two large classes of examples, and you can apply the general name for those classes.

First, for any Lie group $G$ and any closed subgroup $H < G$, the quotient $G/H$ is a manifold, moreover the left action of $G$ on itself descends to a left action on $G/H$, and with respect to this action there is an invariant Riemannian metric (unique up to rescaling). This Riemannian manifold is called a homogeneous space for $G$. In the special case $GL(\mathbb R^2) / O(\mathbb R^2)$, this Riemannian manifold is isometric to the hyperbolic plane $\mathbb H^2$ (up to rescaling).

Second, $GL(\mathbb R^n)$ is a semisimple Lie group. For any semisimple Lie group $G$, up to conjugacy there is a unique maximal compact subgroup $H < G$. The quotient space $G/H$ is a special case of a homogeneous space, called the symmetric space of $G$. For your example, $O(\mathbb R^n)$ is indeed the maximal compact subgroup of $GL(\mathbb R^n)$, and so $GL(\mathbb R^n) / O(\mathbb R^n)$ is the symmetric space of $GL(\mathbb R^n)$.

As it turns out one can also kinda/sorta give a name to the particular symmetric space $GL(n,\mathbb R) / O(n,\mathbb R)$: it is the space of positive definite quadratic forms on $\mathbb R^n$, or if you like the space of positive volume ellipsoids in $\mathbb R^n$, centered on the origin. One way to see this is that $GL(n,\mathbb R)$ acts transitively on this set of ellipsoids, and the stabilizer of the roundest ellipsoid, that is the unit volume round ball centered on the origin, is $O(n,\mathbb R)$.