nLab tensor product of vector bundles

Idea

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.

Definition

Definition

(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

$$\left\{ (g_1)_{i j} \colon U_i \cap U_j \to GL(n_1) \right\} \phantom{AAA} \text{and} \phantom{AAA} \left\{ (g_2)_{i j} \colon U_i \cap U_j \longrightarrow GL(n_2) \right\}$$

be the transition functions of these two bundles with respect to this cover.

For $i, j \in I$ write

$$\array{ (g_1)_{i j} \otimes (g_2)_{i j} &\colon& U_i \cap U_j &\longrightarrow& GL(n_1 \cdot n_2) }$$

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):

$$E_1 \otimes E_2 \;\coloneqq\; \left( \left( \underset{i}{\sqcup} U_i \right) \times \left( \mathbb{R}^{n_1 \cdot n_2} \right) \right)/ \left( \left\{ (g_1)_{i j} \otimes (g_2)_{i j} \right\}_{i,j \in I} \right) \,.$$

Examples

Proposition

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

$$f \;\colon\; E_1 \rightarrow E_2$$

is equivalently a section of the tensor product of $E_2$ with the dual vector bundle of $E_1$:

$$Hom_{Vect(X)}(E_1, E_2) \;\simeq\; \Gamma_X( E_1^\ast \otimes_X E_2) \,.$$

Moreover, this section is a trivializing section (this example) precisely if the corresponding morphism is an isomorphism.

Related concepts

• external tensor product of vector bundles

• direct sum of vector bundles

• inner product on vector bundles

• dual vector bundle

• tensor product

• tensor category

References

• Max Karoubi, §4.8 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226 Springer (1978) [pdf, doi:10.1007%2F978-3-540-79890-3]