Let \(M\) be a smooth manifold and \(E\rightarrow M\) a vector bundle over \(M\).
A connection on the vector bundle \(E\rightarrow M\) is usually defined as a map
\[\nabla : \Gamma(M,TM)\times \Gamma(M,E)\rightarrow \Gamma(M,E)\]
satisfying the following conditions:
- \(\nabla\) behaves very well with the \(\mathbb{R}\)-vector space structure on \(\Gamma(M,TM)\) and \(\Gamma(M,E)\); in the sense that, \(\nabla\) is an \(\mathbb{R}\)-bilinear map,
- \(\nabla\) behaves reasonably well with the \(C^\infty(M)\)-module structure on \(\Gamma(M,TM)\) and \(\Gamma(M,E)\); in the sense that,
\[\nabla(fX,s)=f\nabla(X,s)\] for \(X\in \Gamma(M,TM)\) and \(s\in\Gamma(M,E)\)
\[\nabla(X,fs)=f\nabla (X,s)+X(f)s\] for \(X\in \Gamma(M,TM)\) and \(s\in \Gamma(M,E)\)
The \(C^\infty(M)\)-linearity of the map \(\nabla(-,s):\Gamma(M,TM)\rightarrow \Gamma(M,E)\) for each section \(s\in \Gamma(M,E)\) implies we have an \(\mathbb{R}\)-linear map
\[\Gamma(M,E)\rightarrow \hom_{C^\infty(M)}(\Gamma(M,TM),\Gamma(M,E))\]
We know that \(\hom (V,W)\cong V^*\otimes W\) for vector spaces \(V,W\). More generally, \(\hom_R(V,W)=V^*\otimes_RW\) for \(R\)-modules \(V,W\) for a commutative ring \(R\).
In the case of \(R=C^\infty(M)\), we have
\[\hom_{C^\infty(M)}(\Gamma(M,TM),\Gamma(M,E))\cong \Gamma(M,TM)^*\otimes_{C^\infty(M)}\Gamma(M,E)\],
equivalently,
\[\hom_{C^\infty(M)}(\Gamma(M,TM),\Gamma(M,E))\cong \Gamma(M,T^*M)\otimes_{C^\infty(M)}\Gamma(M,E)\].
We also have a relation between "tensor product of sections" and section of tensor product of vector bundles; given by
\[\Gamma(M,T^*M)\otimes_{C^\infty(M)}\Gamma(M,E)\cong \Gamma(M,T^*M\otimes E)\].
For this reason, some people write a connection on a vector bundle \(E\rightarrow M\) as an \(\mathbb{R}\)-linear map \(\nabla:\Gamma(M,E)\rightarrow \Gamma(M,T^*M\otimes E)\) satisfying the condition
\[\nabla(fs)=(df)\otimes s+f\nabla(s)\]
for all \(f\in C^\infty(M)\) and \(s\in \Gamma(M,E)\).