In differential geometry, a derivation is defined to be an operator $D:C^{\infty}(M)\rightarrow C^{\infty}(M)$ that is linear and satisfies the first-order product rule: $$ D(fg) = D(f)g + fD(g). $$
When I say "second-order derivations," I mean operators of the form $DD'$ where $D, D'$ are derivations. In this case, $DD'$ is linear and satisfies $$ DD'(fg) = DD'(f)g + D(f)D'(g) + D'(f)D(g) + DD'(g). $$
Questions.
- Are there any references that discuss these types of operators?
- Does anyone have an intuitive/geometric interpretation of these operators? First-order derivations can be interpreted as vector fields (or tangent vectors if we're talking about pointwise derivations). Is there an extension of this type of reasoning?
Edit.
It looks like my definition has in general $DD'\ne D'D$. For example, in local coordinates let $D = \partial_{1}$ and $D' = x^{1}\partial_{2}$. Then $$ DD' = \partial_{1}(x^{1}\partial_{2}) = \partial_{2} + x^{1}\partial_{1}\partial_{2} \ne D'D. $$ I'm not sure what definition is appropriate, actually. The above is only my attempt.
Actually, the non-commutativity I just noted above is nothing new. We already know Lie brackets $[X, Y]$ can be nonzero. However, the more I play around, the more surprising nuances I seem to discover.