Conceptual definition of the extension of a connection to 1-forms

Question

I have a question that arose while reading Milnor's "Characteristic Classes". I will use the notation from that book.

Let $M$ be a smooth manifold and let $\zeta$ be a complex vector bundle on $M$. Milnor defines a connection on $M$ to be a map $\nabla\colon C^{\infty}(\zeta) \rightarrow C^{\infty}(\tau_{\mathbb{C}}^{\ast} \otimes \zeta)$ satisfying the Leibniz identity, where $\tau_{\mathbb{C}}$ is the complexified tangent bundle of $M$.

In Lemma 4 of Appendix C, he proves that such a connection can be extended to a map $\hat{\nabla}\colon C^{\infty}(\tau_{\mathbb{C}}^{\ast} \otimes \zeta) \rightarrow C^{\infty}(\wedge^2 \tau_{\mathbb{C}}^{\ast} \otimes \zeta)$ satisfying an appropriate Leibniz rule. However, his proof is just a definition in local coordinates, with the details left to the reader. I verified these details, though they were a bit of a pain.

However, I feel like there must be a more conceptual definition of $\hat{\nabla}$ that makes no reference to local coordinates. Does anyone know one?

Answer 1

If we denote by $\nabla$ the connection on $E\to M$, then we can define an exterior differential $d^\nabla:\Gamma(\Lambda^pM\otimes E)\to\Gamma(\Lambda^{p+1} M\otimes E) $ by $$ d^\nabla \alpha (X_0,\dots, X_p) = \sum_i (-1)^i \nabla_{X_i}(\alpha(\tilde{X_0}, \dots , \hat {\tilde{X_i}}, \dots, \tilde X_p)) + \sum_{i\neq j} -(1)^{i+j} \alpha ([\tilde X_i, \tilde X_j], X_0, \dots, \hat X_i,\dots, \hat X_j, \dots, X_p).$$

where $X_i\in T_x M$; $\tilde X_i$ denotes an extension of $X_i$ to a neighbourhood of $x\in M$, and the hat above something denotes that that argument has been omitted.

This formula can be found in Besse's book "Einstein Manifolds" pg 24, beware I recall there are a few typos in that part of the book.

The pattern for this definition is the usual one used to extend the covariant derivative to the tensor algebra, modified by alternating the result and using the covariant derivative.

Answer 2

Although the answer is already given, it may be helpful to point out how the coordinate definition contains the multilinear definition, in a single step of work. From the definition $$(d^\nabla s)_{ij}^\alpha=\frac{\partial s_j^\alpha}{\partial x^i}-\frac{\partial s_i^\alpha}{\partial x^j}+\Gamma_{i\beta}^\alpha s_j^\beta-\Gamma_{j\beta}^\alpha s_i^\beta$$ you can contract with $X^iY^j$ to get (using product rule from calculus twice) $$(d^\nabla s)(X,Y)^\alpha=\Big(\frac{\partial (s(Y))^\alpha}{\partial x^i}-s_j^\alpha\frac{\partial Y^j}{\partial x^i}\Big)X^i-\Big(\frac{\partial (s(X))^\alpha}{\partial x^j}-s_i^\alpha\frac{\partial X^i}{\partial x^j}\Big)Y^j+\Gamma_{i\beta}^\alpha s(Y)^\beta X^i-\Gamma_{j\beta}^\alpha s(X)^\beta Y^j.$$ The first and fifth terms form the definition of $\nabla_X(s(Y))$, the third and sixth terms form the definition of $\nabla_Y(s(X))$, and the second and fourth terms define $-s([X,Y]).$ So you have $$(d^\nabla s)(X,Y)=\nabla_X(s(Y))-\nabla_Y(s(X))-s([X,Y]).$$