I'm studying differential geometry, and when I began to study connections, the first definition that I found was the following:
Definition: Let M a differentiable manifold. A connection on M is a transformation $\nabla: D(M)\times D(M)\rightarrow D(M)$, where $D(M)$ is the set of differentiable vector fields on $M$, which satisfices:
a)$\nabla_{fX_1+X_2}Y=f\nabla_{X_1}Y+\nabla_{X_2}Y$ with $f\in C^{\infty}(M)$ and $X_1,X_2, Y\in D(M)$
b)$\nabla_X(\lambda Y_1+Y_2)=\lambda\nabla_X Y_1+\nabla_X Y_2$, with $\lambda\in\mathbb{R}$ and $X,Y_1,Y_2\in D(M)$
c)$\nabla_X(fY)=f\nabla_X Y+X(f)Y$, where $f\in C^{\infty}(M)$ and $X,Y\in D(M)$
I had no problem with this definition, but later the book says that we can reinterpret the previous definition, and we can say that a connection is actually a transformation $\nabla:\Gamma(TM)\rightarrow\Gamma(T^*M\otimes TM)$. Explicitly, if $Y\in\Gamma(TM)$, then $\nabla Y$ will be the element of $\Gamma(T^*M\otimes TM)$ which satisfices:
$$\nabla Y(X,\theta)=\theta(\nabla_X Y)$$. So, with this, we can generalize the definition of a connection, but this time in a vector bundle, as follows:
Definition: Let $\xi=(E,\pi)$ a differentiable vector bundle over a differentiable manifold $M$. A connection on $\xi$ is a transformation:
$$\nabla:\Gamma(E)\rightarrow\Gamma(T^*M\otimes E)$$
with the following properties:
a)$\nabla(s)(fX+X',\theta)=f\nabla(s)(X,\theta)+\nabla(s)(X',\theta)$.
b)$\nabla(\lambda s+s')=\lambda\nabla s+\nabla s'$
c)$\nabla(fs)=f\nabla s+df\otimes s$
for all $s,s´\in\Gamma(E)$, $X,X'\in\Gamma(TM)$, $\theta\in\Gamma(E^*)$, $f\in C^{\infty}(M)$ and $\lambda\in\mathbb{R}$.
My problem is that I cannot find the way of joining both definitions. If I take, as particular case, $E=TM$ in the second definition, I don't see why the connection defined in this way is the same (or is connected) with the first one . If the second definition is more general, then it should to reduce to the first one when I take $E=TM$. My little book doesn't explain more, and begins to construct the connection one-forms $\omega_{ij}$. If it is important, my book is "Geometría Riemanniana" of Héctor Sánchez Morgado and Oscar Palmas Velasco.
Best Answer
The thing here is that, to establish the second definition, and write a connection on a manifold $M$ as a transformation $\nabla:\Gamma(TM)\rightarrow\nabla(T^*M\otimes TM)$, it is neccesary to use the isomorphism between $\text{Hom}(TM,TM)$ and $T^*M\otimes TM$ in the following way:
Let $\nabla:\mathfrak{X}(M)\times\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ a connection on $M$ and $Y\in\mathfrak{X}(M)$. If $p\in M$, we define:
$$\nabla Y(p):T_pM\rightarrow T_pM$$
as $(\nabla Y(p))(v):=(\nabla_X Y)(p)$, with $v\in T_pM$ and $X\in\mathfrak{X}(M)$ any differentiable vector field on $M$ such that $X(p)=v$. So, in particular, $\nabla Y(p)$ will be a linear function, and then:
$$\nabla Y(p)\in\text{Hom}(T_pM,T_pM)$$
which means that $\nabla Y\in\Gamma(\text{Hom}(TM,TM))$. But the bundle $\text{Hom}(TM,TM)$ is isomorphic to $T^*M\otimes TM$. This means that we can consider that:
$$\nabla Y\in\Gamma(T^*M\otimes TM)$$
Based on the above, we can reinterpret a connection on $M$ as a transformation:
$$\nabla:\Gamma(TM)\rightarrow\Gamma(T^*M\otimes TM)$$.