I posted this on MSE, but no answer is received, so I post this here.


The problem of finding the $n(n-1)/2$-dimensional volume of the set $SO(n)\subset\mathbb R^{n^2}$ is asked before in this MO [post](https://mathoverflow.net/questions/84848/volume-of-compact-simple-lie-groups-under-the-natural-euclidean-embedding). In the comment section, it is suggested that the answer is $\prod^{n-1}\_{k=1}2^{(k-1)/2}\cdot \textrm{volume}(S^k)$ (where $S^k$ denotes the unit $k$-sphere). It was pointed out that, as I quote: 


"... the factor of $2^{(k-1)/2}$ that you have to put in at each level because the natural map $\pi:SO(k)→S^{k−1}$ is not a Riemannian submersion, but is a Riemannian submersion scaled by a factor of $1/{\sqrt 2}$, i.e., each horizontal vector for this bundle is shrunk by a factor of $\sqrt 2$ by the differential of $\pi$."


My question is:


Explicitly, what is the map $\pi$? How to use $\pi$ to calculate the volume of $SO(n)$? How to compute $d\pi$? Thanks.



Volume of $SO(n)\subset\mathbb R^{n^2}$, again
Maybe this will help: Regard $\mathrm{SO}(n)\subset M\_{n,n}(\mathbb{R})$ as the set of $n$-by-$n$ matrices $a$ that satisfy ${}^ta\,a=\mathrm{I}\_n$ and $\det(a)=1$. Then $\mathrm{SO}(n)$ is a smooth, connected submanifold of $M\_{n,n}(\mathbb{R})$ of dimension $\frac12n(n{-}1)$. 


Give $M\_{n,n}(\mathbb{R})$ the positive definite inner product $\langle x,y\rangle = \mathrm{tr}(\,{}^txy\,)$ for $x,y\in M\_{n,n}(\mathbb{R})$. (See the note at the end about the effects of this choice of inner product.)


Let $E:\mathrm{SO}(n)\to M\_{n,n}(\mathbb{R})$ be the inclusion map, thought of as a $M\_{n,n}(\mathbb{R})$-valued function on $\mathrm{SO}(n)$. Then the induced Riemannian metric on $\mathrm{SO}(n)$ is given by
$$
g = \langle \mathrm{d}E,\mathrm{d}E\rangle = \mathrm{tr}\bigl({}^t(\mathrm{d}E)\,\mathrm{d}E\,\bigr)
= \mathrm{tr}\bigl({}^t(E^{-1}\mathrm{d}E)\,E^{-1}\mathrm{d}E\,\bigr)
= \mathrm{tr}\bigl({}^t\omega\,\omega\,\bigr) 
= - \mathrm{tr}\bigl(\omega\,\omega\,\bigr),
$$
where $\omega = E^{-1}\mathrm{d}E=- {}^t\omega$ is a $1$-form with values in skew-symmetric $n$-by-$n$ matrices, i.e., $\omega = (\omega\_{ij})$ where $\omega\_{ij}=-\omega\_{ji}$. In particular,
$$
g = -\sum\_{i,j}\omega\_{ij}\,\omega\_{ji} = 2\sum\_{i<j} (\omega\_{ij})^2 = \sum\_{i<j} \bigl(\sqrt2\,\omega\_{ij}\bigr)^2.
$$


Meanwhile, writing $E = (e\_1\ e\_2\ \ldots\ e\_n)$ where $e\_i$ is the $i$-th column and hence is a smooth map $e\_i:\mathrm{SO}(n)\to S^{n-1}\subset\mathbb{R}^n$, one has $\mathrm{d}e\_i = e\_j\,\omega\_{ji}$ (sum on $j$) for each $i$. In particular, the pullback of the metric on $S^{n-1}$ by the submersion $e\_n:\mathrm{SO}(n)\to S^{n-1}$ is the quadratic form 
$$
g\_n = \mathrm{d}e\_n\cdot \mathrm{d}e\_n = (e\_j\,\omega\_{jn})\cdot (e\_i\,\omega\_{in}) = \sum\_{i<n} (\omega\_{in})^2.
$$
Thus, $e\_n:\mathrm{SO}(n)\to S^{n-1}$ is a Riemannian submersion (with fibers isometric to $\mathrm{SO}(n{-}1)$) only after scaling the metric $g$ by a factor of $1/2$.


Consequently, we have the formula
$$
\mathrm{vol}\bigl(\mathrm{SO}(n)\bigr) = (\sqrt 2)^{n-1}\mathrm{vol}(S^{n-1}) \mathrm{vol}\bigl(\mathrm{SO}(n{-}1)\bigr),
$$
which, by induction, yields
$$
\mathrm{vol}\bigl(\mathrm{SO}(n)\bigr) 
= (\sqrt 2)^{n(n-1)/2}\,\mathrm{vol}(S^{n-1})\mathrm{vol}(S^{n-2})\cdots
\mathrm{vol}(S^{1}).
$$


**N.B.** If one doesn't like the factors of $\sqrt 2$ that appear in this formula, one can fix this problem by instead taking the metric on $M\_{n,n}(\mathbb{R})$ to be the positive definite inner product $\langle x,y\rangle = \tfrac12\,\mathrm{tr}(\,{}^txy\,)$ for $x,y\in M\_{n,n}(\mathbb{R})$. In that case, the $\sqrt2$ factors go away. Some sources use this convention, others do not. If this is important in one's calculations, one should always check the convention of the source.


