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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?

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closed as not a real question by Steven Landsburg, Andreas Blass, Deane Yang, Misha, Steven Sam Mar 22 '13 at 15:55

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

    
First, how is your actual question related to connections? Secondly, on any bundle, you can construct a section by patching it together locally with partitions of unity! –  Kofi Mar 22 '13 at 8:42
    
I am not sure whether you want to define a connection on the tensor product, or prove the existence of a smooth section of the tensor product, or construct a smooth section out of smooth sections of the factor bundles of the tensor product, or define what the expression ``smooth section'' means for the tensor product bundle (given that you know what it means for the two factor bundles, I suppose). –  Ben McKay Mar 22 '13 at 12:07
    
I just want to know how we can define a sections of a tensor product of two vector bundles . –  DAVID Mar 22 '13 at 12:20
    
I can't tell whether you're asking for the definition of a section or for a method of constructing sections. –  Steven Landsburg Mar 22 '13 at 12:53
    
the method we construct the sections of the tensor product of two vector bundles. –  DAVID Mar 22 '13 at 13:14
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1 Answer 1

up vote 2 down vote accepted

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.

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