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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.

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  • $\begingroup$ How about the case $n=1$ ? $\endgroup$
    – hardmath
    May 18 at 19:48
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    $\begingroup$ Polar decomposition does it. $\endgroup$
    – Ruy
    May 18 at 20:11
  • $\begingroup$ By the QR decomposition (or RQ decomposition if you want to consider the right action of $O(n)$ on $GL(\mathbb{R}^n)$) the quotient space is in bijection with the set of upper-triangular matrices with positive terms on diagonal. Set of such matrices is just $(\mathbb{R}_{> 0})^n \times (\mathbb{R})^{\frac{n(n-1)}{2}} \cong \mathbb{R}^{\frac{n(n+1)}{2}}$. $\endgroup$
    – pmp
    May 18 at 20:30

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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 normalized positive definite quadratic forms on $\mathbb R^n$, or if you like the space of volume 1 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 $O(n,\mathbb R)$.

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    $\begingroup$ Quibbles: of course $GL_2(\mathbb R)$ is officially reductive, not semi-simple. Also, it's $SL_2(\mathbb R)/SO(2,\mathbb R)$ that is isomorphic to the upper half-plane. Yes, $GL_w$ contains the slightly larger $O(2,\mathbb R)$, but/and the quotient is the Cartesian product of the upper half-plane with a ray (isomorphic to the positive diagonal matrices...) $\endgroup$ May 18 at 21:16
  • $\begingroup$ Yes, I have to admit to lack of precise knowledge when it comes to things like this, being more of a geometric group theorist than a Lie groupie. $\endgroup$
    – Lee Mosher
    May 19 at 1:13

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