About: Tensor product bundle

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In differential geometry, the tensor product of vector bundles E, F (over same space ) is a vector bundle, denoted by E ⊗ F, whose fiber over a point is the tensor product of vector spaces Ex ⊗ Fx. Example: If O is a trivial line bundle, then E ⊗ O = E for any E. Example: E ⊗ E ∗ is canonically isomorphic to the endomorphism bundle End(E), where E ∗ is the dual bundle of E.

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  • In differential geometry, the tensor product of vector bundles E, F (over same space ) is a vector bundle, denoted by E ⊗ F, whose fiber over a point is the tensor product of vector spaces Ex ⊗ Fx. Example: If O is a trivial line bundle, then E ⊗ O = E for any E. Example: E ⊗ E ∗ is canonically isomorphic to the endomorphism bundle End(E), where E ∗ is the dual bundle of E. Example: A line bundle L has tensor inverse: in fact, L ⊗ L ∗ is (isomorphic to) a trivial bundle by the previous example, as End(L) is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space X forms an abelian group called the Picard group of X. (en)
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  • In differential geometry, the tensor product of vector bundles E, F (over same space ) is a vector bundle, denoted by E ⊗ F, whose fiber over a point is the tensor product of vector spaces Ex ⊗ Fx. Example: If O is a trivial line bundle, then E ⊗ O = E for any E. Example: E ⊗ E ∗ is canonically isomorphic to the endomorphism bundle End(E), where E ∗ is the dual bundle of E. (en)
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  • Tensor product bundle (en)
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