The nonvanishing top-degree form $\omega^m$ is a <volume form>. Declare an ordered tangent <basis> positive when its value under $\omega^m$ is positive. Smooth nonvanishing ensures these choices agree on overlapping oriented charts and vary continuously. Hence \b[a <symplectic manifold> is orientable]; this is its symplectic <orientation>.

Solution

Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / B / 6 / i / Solution

 Solution (source code)