Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.

Definition:Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative if $\forall X,Y,Z \in \mathfrak{X}(M)$: $\nabla_{\nabla_X Y}Z = \nabla_X\nabla_Y Z$.

Using the definitions of the tensors $R$ and $T$, it is immediate that a necessary condition for $\nabla$ to be 'associative' is $$ R(X,Y)Z = \nabla_{T(X,Y)}Z $$

The motivation for this definition is the single curious observation that if $\nabla$ is 'associative', then $\mathfrak{X}(M)$ becomes an *associative* $\mathbb{R}$-algebra via the multiplication $X\ast Y:=\nabla_X Y$, although at the moment I am not really sure what this is good for... But one step at a time!

I've added the complex geometry tag, because I am actually more interested in the complex setting (i.e. associative $\mathbb{C}$-algebras). In other words, I look at such a pair $(M,\nabla)$, if existing, as another source for associative (complex, possibly Banach?) algebras and would like to know how sound the differential-geometric side of it is.