Isomorphism of tangent and cotangent spaces induced by a symplectic structure on a manifold

Question

If $(M^{2n}, \omega^2)$ is a symplectic manifold, then for each tangent vector $v \in T_xM$ for some $x \in M$ we may associate a $1$-form $\omega_v^1 \in T_x^* M$ by $\omega_v^1(u) = \omega^2(u,v)$.

The map carrying a vector to it's associated $1$-form is an isomorphism. My issue is with proving this.

Showing that it is injective is simple enough by looking at the kernal. It is surjectivity that I am stuck on. I'm trying to show that given an arbitrary $1$-form on $T_x M$, there is a vector, such that the image of this vector under the map constructed above, is the form.

You shouldn't edit the title of the post to mark it as solved, instead accept the answer giving the solution. — Mark Schultz-Wu, Dec 29 '17 at 02:48
By the way this also gives you an isomorphism between the tangent bundle and the cotangent bundle — Exit path, Dec 29 '17 at 08:45

score 4 · Accepted Answer · answered Dec 29 '17 at 02:45

4

If you can show that it is injective, then it is surjective since both spaces have the same dimension.

answered Dec 29 '17 at 02:45

Tsemo Aristide

87,475

Isomorphism of tangent and cotangent spaces induced by a symplectic structure on a manifold

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