I'd like to know if there is a higher order (third order, fourth order, ...) analogues of Riemann curvature tensor or other related notions of curvature on vector bundles.
Background
Given a smooth vector bundle $E \rightarrow M$, one way to define a connection on it is to give a linear map: $$\nabla: \Gamma(M,\wedge^p T^*M \otimes E) \rightarrow \Gamma(M, \wedge^{p+1} T^*M \otimes E)$$ such that for $p=0$, $\nabla(\varphi \xi) = d \varphi \cdot \xi + \varphi \cdot \nabla \xi$ for each smooth function $\varphi$ and $\xi \in \Gamma(M,E)$, and extend this notion to higher order $E$-valued forms.
A more down-to-earth notion of directional derivative can be obtained from the above definition as: $$\nabla_X \xi = \langle \xi, \nabla X\rangle $$ for $\xi \in \Gamma(M,E)$ and $\langle \cdot , \cdot \rangle$ being contraction of a covector with a vector.
From a connection $\nabla$, we get $$\nabla^2 := \nabla \circ \nabla : \Gamma(M,E) \rightarrow \Gamma(M, \wedge^2 T^*M \otimes E)$$ which we may define as curvature because for $E = TM$, it satisfies the following identity: $$(\nabla^2 Z)(X,Y) = R(X,Y)Z = ([\nabla_X, \nabla_Y] - \nabla_{[X,Y]})Z$$ Therefore studying Riemann curvature tensor $R$ is equivalent to studying $\nabla^2$.
The obvious generalization is to ask: can we ascribe meaning to $\nabla^3, \nabla^4, \cdots$, and associate meaningful quantities to them? For example, we may ask if there is an expression of the form: $$(\nabla^k Y)(X_1, \cdots X_k) = \sum_i P_i(\nabla_{Q_{i,1}(X_1, \cdots X_k)}, \cdots \nabla_{Q_{i,m}(X_1, \cdots X_k)})Y$$ where $P,Q$ are (noncommutative) polynomials.
Note: this question is similar to this question, which didn't have any answer.
