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I wonder why tangent bundle is not symplectic. 

As you know, cotangent bundle is symplectic.

(1) question 1 : Is cotangent bundle isometric to tangent bundle ?

(2) question 2 : Why is not tangent bundle symplectic manifold ?

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up vote 8 down vote accepted

The tangent bundle and cotangent bundle of any manifold $M$ are always isomorphic as bundles. (Note that it doesn't make sense to talk about isometric until a metric is chosen).

To see they are diffeomorphic, choose a Riemannian metric $g$ on $M$. Define a map $TM\rightarrow T^\ast M$ by sending $(p,v) \rightarrow g_p(v,\cdot)$. One easily verifies this is a bundle isomorphism.

In particular, the abstract manifolds $TM$ and $T^\ast M$ are diffeomorphic. This implies, by transport of structure, that $TM$ can always be given the structure of a symplectic manifold. The issue is that while $T^*M$ is canonically a symplectic manifold, $TM$ is not - the symplectic structure inherited from $T^\ast M$ depends on the choice of $g$.

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This is a great answer ! – Damien L Apr 11 '13 at 23:17

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