# Is the exponential map a symplectomorphism?

In the case of $\mathbb{R}^{2n}$ the exponential map just becomes $$exp_{p}(v) = p + v$$ which results in $$Dexp_{p}(w)_{v} = w$$ and proves that this map is a symplectomorphism.

I am wondering if this is also the case in a general symplectic manifold and in the likely event that it isn't true what possible properties are necessary to prove it is a symplectomorphism.

• see Lectures on Symplectic Geometry-Ana Cannas da Silva 4.2 – Bey Alexander Oct 30 '18 at 19:20
• @BeyAlexander The mentioned reference gives a property for a general symplectomorphism, I am wondering if their is one more specificly for exponential maps? – Netivolu Oct 30 '18 at 19:50