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I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following:

If $B$ is a zero neighborhood in Banach space. Let $\Omega$ be a symplectic form on this banach space. Let $\Omega_1$ be the symplectic structure on $B$ which is constant with respect to the natural parallelism on $B$ and equal to $\Omega$, at $0$ then...

I want to understand the meaning of natural parallelism here.

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up vote 3 down vote accepted

A parallelism (or parallelization) on a smooth manifold $M,$ modelled on a Banach space $\mathbb E,$ is determined by a global trivializing chart $\phi:TM\to M\times\mathbb E$ of its tangent bundle $\tau_M:v_p\in TM\mapsto p\in M.$

A smooth differential $k$-form $\omega$ on $M,$ is said to be constant w.r.t. the given parallelism when there exists $F,$ a bounded multilinear (alternating) $k$-form on $\mathbb E,$ such that $$\omega_{p}(v_1,\dots,v_k)=F(\phi|_p v_1,\dots,\phi|_p v_k),$$ for any $p\in M,$ and $v_1\dots,v_k\in T_pM.$

The natural parallelism of an open subset $U$ of $\mathbb E$ is the identity map of $TU=U\times\mathbb E.$

Note that, if there exists a parallelism on $M,$ then $M$ is called parallelizable.

Edit As a reference you can look at the entry "Parallelism" (originally written by D.Alekseevskii) in the EOM. Even if there are considered only finite-dimensional manifolds, the generalization is obvious.

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Thnks for the answer. – zapkm Aug 28 '12 at 4:47

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