Suppose we have some smooth manifold with some connection $\nabla$. I understand how $\nabla$ acts on vectors ($\nabla_{e_i} e_j$) and tensors ($\nabla_{e_i} (e_j \otimes e_k \otimes \dots)$). Those are the two applications with an obvious geometric interpretation.
Can this notion be extended to covariant differentiation by tensors? I.e. $\nabla_{e_i \otimes e_j \dots} (e_k)$? I'm not sure what the geometric picture would be.
I've been reading through Clifford Algebra to Geometric Calculus (Hestenes) and they introduce the idea of a multi-vector derivative. I've been trying to understand this idea by working in a given basis, but I'm confused about how to handle expressions like the one above. For functions of multi-vector valued variables the multi-vector derivative is easy, but in the more general case, I'm not sure how to proceed. Can someone provide some clarification?
(Aside: I know tensors are not one-to-one with multi-vectors, but they represent similar constructions. The geometric algebra is a quotient of the tensor algebra so the logic should extend)