Levi-Civita connection on forms

Question

I know that on a Riemannian manifold $(M,g)$, one can extend the Levi-Civita connection on one-forms. However I only know how to extend a connection on tensor products. Is it possible to extend the Levi-Civita connection to any form and if yes how?

Answer 1

Short answer to your question is Yes. The way to do it is by "stealing the algebraic formula of Leibniz rule". Let's begin by recalling the "Tensor Characterization Lemma".

Tensor characterization Lemma: Let $M$ be a manifold, let $F:\chi(M)^k\rightarrow C^{\infty}(M)$ be a $C^{\infty}(M)$ multilinear map. Then there exists a unique smooth $(k,0)$ tensor field $w$ such that for every $X_1,X_2,...,X_k \in C^{\infty}(M)$ and every $p\in M$ we have that $F(X_1,X_2,..,X_k)|_p=w|_p(X_1(p),X_2(p),...,X_k(p))$.

The above lemma allows us to naturally identify smooth tensor fields with $C^{\infty}(M)$ multilinear map. This is a convineant algebraic point of view. Once we are done with this identification, we just need to define $C^{\infty}(M)$ multilinear map $\nabla_YF$ in $k$ arguments for any $C^{\infty}(M)$ multilinear map $F$ in $k$ arguments and any vector field $Y$ as below:

$$(\nabla_YF)(X_1,X_2,...,X_k)= Y(F(X_1,X_2,...,X_k))-\sum_{i=1}^k F( (\nabla_Y)^i(X_1,X_2,...,X_k)) $$

Where $X_1,X_2,...,X_k$ are arbitrary vector fields and $(\nabla_Y)^i(X_1,X_2,...,X_k)$ means apply the operator $\nabla_Y$ to $X_i$. One checks easily that $\nabla_YF$ will be indeed $C^{\infty}(M)$ multilinear map in $k$ arguments.

The above was an algebraic approach, here is an alternative geometric approach:

Let $w$ be smooth $(k,0)$ Tensor field and let $(p,v)\in TM$, then we define the multilinear map $\nabla_{(p,v)}w:(T_pM)^k\rightarrow \mathbb{R}$ as follows:

Let $\gamma$ be any smooth path such that $\gamma'(0)=(p,v)$, then for any tangents $u_1,u_2,...,u_k$ at $p$ we define $\nabla_{(p,v)}w(u_1,u_2,...,u_k)$ to be $\frac{d}{dt}|_0(w|_{\gamma(t)}(P^t(u_1),P^t(u_2),...,P^t(u_k))$, where $P^t:T_pM\rightarrow T_{\gamma(t)}M$ is the parallel transport map along $\gamma$. It turns out that this definition is independent of the choice of the smooth path $\gamma$ and so is a valid way to define $\nabla_{(p,v)}w$