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Given two vector bundles $V_1 \to X$ and $V_2 \to X$ over the same base space $X$, their tensor product of vector bundles $V_1 \otimes V_2 \to X$ is the vector bundle over $X$ whose fiber over any point is the tensor product of vector spaces (i.e. the tensor product of modules) of the respective fibers of $V_1$ and $V_2$ (the fiber-wise tensor product).
The tensor product of vector bundles makes the category Vect(X) into a symmetric monoidal category, in fact a distributive monoidal category.
(tensor product of topological vector bundles)
Let $X$ be a topological space, and let $E_1 \to X$ and $E_2 \to X$ be two topological vector bundles over $X$.
Let $\{U_i \subset X\}_{i \in I}$ be an open cover with respect to which both vector bundles locally trivialize (this always exists: pick a local trivialization of either bundle and form the joint refinement of the respective open covers by intersection of their patches). Let
be the transition functions of these two bundles with respect to this cover.
For $i, j \in I$ write
be the pointwise tensor product of vector spaces of these transition functions
Then the tensor product bundle $E_1 \otimes E_2$ is the one glued from this tensor product of the transition functions (by this construction):
Let $X$ be a topological space and let $E_i \overset{p_i}{\to} X$ be a two topological vector bundles over $X$, of finite rank of a vector bundle. Then a homomorphism of vector bundles
is equivalently a section of the tensor product of $E_2$ with the dual vector bundle of $E_1$:
Moreover, this section is a trivializing section (this example) precisely if the corresponding morphism is an isomorphism.
Last revised on May 17, 2023 at 10:07:27. See the history of this page for a list of all contributions to it.