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I recently read the principal $G$-bundle construction on a smooth manifold $M$, where $G$ is a Lie group.

To understand them better, I am looking for some applications.

Can the principal $G$-bundle help us get some usual bundle constructions, for example tensor product of two vector bundles, the pullback bundle etc?

Right now, the constructions I have seen are specific to each type of construction. If I want the tensor product of two vector bundles $E$ and $F$ over a smooth manifold $M$, I start from scratch and consider the disjoint union $\bigsqcup_{p\in M} E_p\otimes F_p$ and put trivializations "naturally". Similarly for the dual bundle.

Is there a unified way to think of these constructions so that all the constructions are dealt with in one shot?

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    $\begingroup$ I think what you want to look into is associated bundles of the frame bundle. As I recall this is the way to naturally generate all manner of tensors. But, I think the frame bundle is also a principal bundle with respect to $G = GL(n)$ so, my comment is not so far off your idea... This math.stackexchange.com/questions/649142/… might be helpful, more to the point see en.wikipedia.org/wiki/Frame_bundle $\endgroup$ Jun 17, 2015 at 17:23

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If $f : G \to H$ is a morphism of Lie groups, it induces a functor from principal $G$-bundles to principal $H$-bundles (explicitly, apply $f$ to Cech cocycles). This construction is itself functorial. This subsumes the usual bundle constructions. For example:

  • Direct sum comes from morphisms $GL_n \times GL_m \to GL_{n+m}$.
  • Tensor product comes from morphisms $GL_n \times GL_m \to GL_{nm}$.
  • Dual comes from the inverse transpose morphisms $GL_n \to GL_n$.
  • "Underlying real bundle" comes from morphisms $GL_n(\mathbb{C}) \to GL_{2n}(\mathbb{R})$.
  • "Complexification" comes from morphisms $GL_n(\mathbb{R}) \to GL_n(\mathbb{C})$.

And so forth.

But the idea of principal bundles has many other applications. Some have a more homotopy-theoretic flavor, e.g. the theory of characteristic classes and reduction of the structure group, while others have a more differential-geometric flavor, e.g. connections and Chern-Weil theory.

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  • $\begingroup$ This is great. Will need some time to digest. $\endgroup$ Jun 17, 2015 at 18:27
  • $\begingroup$ If you prefer to work globally rather than with Cech cocycles, the functor from principal G-bundles to principal H-bundles can also be described by the "extension of scalars" $P \mapsto P \times_G H$. $\endgroup$
    – ಠ_ಠ
    Feb 2, 2017 at 1:00

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