I am trying to connect two different notions of connection I learned:
I first learned the notion of connections in Demailly's "Complex Analytic and Differential Geometry", where in the section called "Linear Connections", he defined a connection $D$ on a bundle $E$ as
a linear differential operator of order 1 acting on $\mathcal{C}_*^\infty(M,E)$ and satisfying the following properties:
$$\text{(i)} D:\mathcal{C}_q^\infty(M,E)\to \mathcal{C}_{q+1}^\infty(M,E)$$
$$\text{(ii)} D(f\wedge s)=df\wedge s+(-1)^q f\wedge Ds$$ for any $f\in \mathcal{C}_p^\infty(M,K)$ and $s\in \mathcal{C}_q^\infty(M,E)$.
Here he appears to use $\mathcal{C}_q^\infty(M,E)$ to denote the bundle $\bigwedge^q T^*M\otimes E$
At the start of "Elliptic operators, topology and asymptotic methods", the author defines a connection on a vector bundle $V$ as
a linear map $$\nabla: C^\infty(TM)\otimes C^\infty(V)\to C^\infty(V)$$ assigning to a vector field $X$ and a section $Y$ of $V$ a new vector field $\nabla_X Y$ such that, for any smooth function $f$ on $M$,
$$\text{(i) }\nabla_{fX} Y=f\nabla_X Y$$
$$\text{(ii) }\nabla_X(fY)=f\nabla_X Y+(X.f)Y,$$ where $X.f$ denotes the Lie derivatives of $f$ along $X$.
Are these two notions of connection the same? Would $\nabla_X\nabla_Y(s)$ be the same as $D^2(s)$ evaluated on $(X,Y)$? (I am not sure how to evaluate a section of $\bigwedge^2 T^*M\otimes E$ on a pair of vector fields, so this is my best guess of how could these two be equivalent.)
Best Answer
The first definition sounds much more like the exterior covariant derivative associated to a linear connection (as in the second definition); another common notation for it is $d^{\nabla}$ or $d_{\nabla}$. Also, in the second definition, you write
the second instance of vector field (emphasis mine) is a typo, it should say "a section $\nabla_XY$ of $V$".
Also, in the first definition, $D^2s$, the second covariant exterior differential of a smooth section $E$-valued $q$ form $s$ on $M$, equals the 'wedge' of the curvature, $R$, of the connection (which is a smooth $\text{End}(E)$-valued $2$-form on $M$, i.e $R\in \mathcal{C}^{\infty}_2(M,\text{End}(E))$) and $s$: \begin{align} D^2s&=R\wedge_{\epsilon}s. \end{align} Here, $\wedge_{\epsilon}$ denotes the wedge product with respect to evaluation of endomorphisms on vectors: see Ivo Terek's answer here for details. Its value on a pair of vector fields $X,Y$ is \begin{align} (D^2s)(X,Y)&=\nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s. \end{align}
As far as terminology is concerned, I find this usage confusing, but the operators are essentially the same when acting on sections of $E$, i.e $D:\mathcal{C}^{\infty}_0(M,E)\to \mathcal{C}^{\infty}_1(M,E)$ and $\nabla:C^{\infty}(TM)\otimes C^{\infty}(E)\to C^{\infty}(E)$ are related as: \begin{align} (Ds)(X)&=\nabla_Xs. \end{align} Or, $Ds=\nabla s$, where it is understood that there is an open slot to be filled in by a vector field $(Ds)(\cdot)=\nabla_{(\cdot)}s$; this open slot for a vector field is what gives the 1-form nature.