When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
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$\begingroup$ What is the definition of real polarization in this setting? Usually I have seen that it is a Lagrangian fibration satisfying some properties, but that doesn't fit with what you've written. $\endgroup$– user36931Commented Dec 25, 2013 at 16:45
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$\begingroup$ here is the definition of real polarization in sense of geometric quantization maths.ed.ac.uk/~jthomas7/GeomQuant/Lecture4.pdf $\endgroup$– user21574Commented Dec 25, 2013 at 16:47
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1 Answer
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Well, this only would make sense when $n$ is even, but there are two problems: First, except when $n=1$, the $2n$-sphere does not carry any symplectic structure. Second, the tangent bundle of the $2n$-sphere has no nontrivial subbundles anyway. (The reason is that $TS^{2n}$ has nonzero Euler class, so it cannot be written as $P\oplus Q$ where $P$ and $Q$ have positive rank, since these two bundles would have to be orientable and have zero Euler class.)
answered Dec 25, 2013 at 17:47
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$\begingroup$ Thanks a lot, can you explain more the last two lines, it is a bit mass for me. :) $\endgroup$– user21574Commented Dec 25, 2013 at 18:06
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1$\begingroup$ If you don't know about the Euler class of an oriented vector bundle, I suggest that you take a look at the chapter on this in Milnor and Stasheff's Characteristic Classes, where the Euler class and the Euler product formula are explained clearly and simply. $\endgroup$ Commented Dec 25, 2013 at 19:23
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$\begingroup$ Thanks a lot. I am stadying this book now. Happy new year $\endgroup$– user21574Commented Dec 25, 2013 at 20:26