Riemannian curvature tensor and Schwarz theorem

Question

In multivariable calculus we have Schwarz Theorem that states that partial derivatives commute. In Riemannian geometry we have the curvature tensor which can be seen as a measure for how much the derivatives do not commute. I would like to know some examples (probably there are very easy results) that work in multivariable calculus thanks to Schwarz Theorem which fail in Riemannian geometry.

Answer 1

Not quite an answer but in Riemannian geometry if you work in a local chart then partial derivatives do commute. It is the covariant derivatives (where the Christophel symbols enter and which are 'independent' of charts) that do not necessarily commute. You may interpret partial derivatives as covariant derivatives by putting a flat metric on the plane. Then the Christophel symbols vanish and you are just left with the partial derivative part.).