If $f$ is a smooth function on a symplectic manifold $(M, \omega)$ we can define its symplectic gradient : this is a vector field $X_f$ such that $\iota_{X_f} \omega = - df$. My question is the following : consider a critical point of $f$, does it makes sense to talk about "symplectic Hessian" at this point ? If yes, how can we define/compute it ?
Symplectic gradient
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differential-geometry
symplectic-geometry
morse-theory
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4Having used the symplectic structure to obtain a vector field, you can compute its intrinsic derivative at a zero. See [this](http://math.stackexchange.com/questions/1462819/what-is-the-meaning-of-dx-p-for-x-a-vector-field-on-a-manifold/1462925#1462925). – 2017-01-11
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0This is great, thanks a lot ! – 2017-01-11
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2You're most welcome. – 2017-01-11