I would be pleased to have some information about an alternate definition for the torsion tensor.

Let us consider a smooth manifold $\mathcal{M}$ together with an arbitrary connection $\nabla$. The first definition for the torsion that I mostly see in the (physics) literature is the following:

${\bf T} ({\bf e}_\alpha, {\bf e}_\beta) := {\nabla}_{{\bf e}_\alpha} {\bf e}_\beta - {\nabla}_{{\bf e}_\beta} {\bf e}_\alpha - [{\bf e}_\alpha,{\bf e}_\beta] \, , $

with $\{ {\bf e}_\alpha \}$ being an arbitrary basis on $\mathcal{M}$. The other definition, always encountered in terms of components in the literature that I have read, is, in its tensorial form,

${\bf T} ({\bf e}_\alpha, {\bf e}_\beta)(f) := \nabla \nabla f ({\bf e}_\alpha,{\bf e}_\beta) - \nabla \nabla f ({\bf e}_\beta,{\bf e}_\alpha) \, , $

where $f$ is a smooth function on $\mathcal{M}$. One can readily check that, for instance in terms of the connection coefficients of $\nabla$, these two definitions are equivalent.

My questions are

(i) As I did not find the second definition in the few mathematics books that I have read, is it indeed a correct definition? (I realize that this question can sound strange as both definitions yield the same result, but not having seen it in a book yet makes me wonder.)

(ii) What's the meaning of the second definition?

(iii) Can the second definition be written in terms of an exterior derivative, and if so, what would be the form?

Thanks in advance for your help.

Xavier

Morse Theory, Definition 8.5 and the attached footnote. – Robert Bryant Jan 24 at 12:20