Suppose we have a smooth manifold M and E--->M is a vector bundle. A connection on E is a linear map from the set of all smooth section on E into the set of smooth sections of the tensor product of E and the cotangent bundle of M , satisfying a condition.
Here is the question : 
We can make the tensor product bundle of two vector bundles on a same base space which is here the smooth manifold , M , . But how can we define a section on this new vector bundle?



sections of tensor product bundle ( tensor product of two vector bundles )
If $s$ is a smooth section of a vector bundle $V$ and $t$ is a smooth section of a vector bundle $W$ then $s \otimes t$ is a smooth section of $V \otimes W$. Its value at each point $m \in M$ in the underlying manifold $M$ is $(s \otimes t)(m)=s(m) \otimes t(m)$. I am still not sure if that is what you are looking for. This has nothing to do with connections, so I don't see why you mention connections in your question. If $s\_1, s\_2, \dots, s\_p$ is a basis of local smooth sections of $V$ over an open set $U$ (i.e. these sections are defined on $U \subset M$ and every local smooth section defined on $U$ is a unique linear combination $\sum\_i f\_i s\_i$ with $f\_i$ smooth functions) and similarly $t\_1, t\_2, \dots, t\_q$ is a basis of local sections of $W$ over the same open set $U$, then $s\_i \otimes t\_j$ for $i=1,2,\dots,p$ and $j=1,2,\dots,q$ is a basis of local smooth sections of $V \otimes W$ over $U$. So that should explain what all of the sections look like, I hope.


