# Volume form on real Grassmanian

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?

• 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. – Ben McKay Dec 27 '16 at 20:05
• 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 – Ben McKay Dec 27 '16 at 20:07
• Thank you! I guess, it gives positive answer to my question. – Daniil Rudenko Dec 27 '16 at 21:29