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It's well known that the even-dimensional complex quadric $Q_{2n}$, defined by the equation $z_1^2+\cdots +z_{2n+2}^2=0$ in complex projective space $CP^{2n+1}$, is diffeomorphic with the Grassmann manifold of oriented real 2-planes. A natural diifeomorphism is given as follows: for any two orthogonal vectors $v_1$ and $v_2$ in $R^{2n+2}$, the oriented 2-plane spanned by $v_1, v_2$ corresponds to $\pi(v_1+iv_2)$ where $\pi$ is the natural projection $C^{2n+2}\backslash \{0\}\rightarrow CP^{2n+1}$.

My question is: what part of this breaks down for the odd-dimensional quadrics?

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I don't see a problem. A quadric in $\mathbb CP^2$ is topologically a sphere, which paramaterizes oriented real $2$-planes in $\mathbb R^3$. – Will Sawin Dec 22 '12 at 1:34
@Sawin: You're right, there's no problem. I was reading a paper that wasn't worded properly; the author gave me the misleading impression the result was only true in even dimensions. This question can be deleted. – Oliver Jones Dec 22 '12 at 2:22
Don't delete it! I learned something from it. – Gene Ward Smith Dec 23 '12 at 3:54

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