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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?

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

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