There exists a unique $SO(4)-$invariant volume form on the real Grassmanian G(2,2). Is it possible to write it down algebraically in Plucker coordinates?
$\begingroup$
$\endgroup$
3
asked Dec 27, 2016 at 18:27
-
4$\begingroup$ Careful: to get the Grassmannian to be oriented, you need it to be the Grassmannian of oriented 2-planes in $\mathbb{R}^4$. The volume form is then unique up to rescaling by a nonzero constant. $\endgroup$– Ben McKayCommented Dec 27, 2016 at 20:05
-
6$\begingroup$ In Plucker coordinates, the oriented Grassmannian is a product $S^2 \times S^2$ and the volume form is the product volume form, with equal area on each sphere. See my thesis arxiv.org/abs/math/0101017 $\endgroup$– Ben McKayCommented Dec 27, 2016 at 20:07
-
$\begingroup$ Thank you! I guess, it gives positive answer to my question. $\endgroup$– Daniil RudenkoCommented Dec 27, 2016 at 21:29
Add a comment
|