How to prove that the Grassmannian of oriented subspaces $\mathrm{Gr}_+(2,4,\mathbb R)$ is homeomorphic to $S^2\times S^2$? I know that $\mathrm{Gr}_+(2,4,\mathbb R)\cong\mathrm{SO}(4)/(\mathrm{SO}(2)\times\mathrm{SO}(2))$, but don't know whether this can help.
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1http://people.math.harvard.edu/~klin/Gr24.pdf – Thomas Rot Jan 20 '20 at 15:17
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It is also in Besse's book "manifolds all of whose geodesics are closed." – Thomas Rot Jan 21 '20 at 08:46