Hello


I was wondering if anybody can direct me to a paper or a book regarding the volume of $Gr(2,4) $ or generic complex Grassmanian manifolds of order $k$. My own heuristic method seems not to work! It is based on the adaption of the same procedure one has to follow for finding the volume of complex projective spaces $\mathbb{C}P^n$ using Hopf fibration $\mathbb{C}P^n\cong S^{2n+1}/S^1$. Here the volume can be roughly given by dividing the volume of $2n+1-$sphere by volume of $S^1$. Therefore in analogy with this example, we can estimate the volume of $Gr(k,n)$ by dividing the volume of $U(n)$ by that of $U(n-k) \times U(k)$ which gives me $12\pi^4 r^{16}$ for $Gr(2,4)$ where $r$ is the radius of $S^1$ and I don't like it because $Gr(2,4) $ is $8$ dimensional! 


Thanks in Advance
AB



Volume of Gr(2,4)
Check section 9.1.2 of [these notes](http://www3.nd.edu/~lnicolae/Lectures.pdf) There I compute the volumes of *real* Grassmannians. A similar computation works in the complex case.


**Update** Using the description $\mathrm{Gr}\;(k, N)\cong U(N/U(k)\times U(N-k)$ and a bi-invariant metric on $U(N)$, this induces bi-invaraint metrics on $U(k),U(N-k)\subset U(n)$ and an invariant metric on $\mathrm{Gr}(k,N)$. The volume of $\mathrm{Gr}(k,N)$ with respect to this metric is


$$ {\rm vol} \mathrm{Gr}(k, N)= \frac{ {\rm vol}\; U(N)}{{\rm vol}\; U(k)\cdot {\rm vol}\; U(N-k)}. $$


The volume of a compact Lie group $G$ with respect to a bi-invariant metric $g$ was computed by I.G. Macdonald, 



> 
> *The volume of a compact Lie group*, Invent. Math. **56**(1980), no. 2, 93–95.
> 
> 
> 


For the Lie group $U(n)$ this takes the form


$$ {\rm vol}\; U(n)=\frac{1}{(2P\_n)^2(2\pi)^n}\times {\rm vol}\; T^n\times \prod\_{k=1}^n {\rm vol}\;S^{2k-1}, $$


where ${\rm vol}\; T^n$ denotes the volume of the maximal torus of $U(n)$ equipped with the induced bi-invariant metric, and $P\_n$ is the product of the lengths of the positive roots of $U(n)$.


