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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?

- Basis of cotangent space
- Geometric meaning of symmetric connection
- Prove that a straight line is the shortest distance between two points?
- Antipodal mapping of the sphere
- Distance between two points on the Clifford torus
- What exactly is a Kähler Manifold?
- Orientation preserving diffeomorphism.
- Why is the geometric locus of points equidistant to two other points in a two-dimensional Riemannian manifold a geodesic?
- On surjectivity of exponential map for Lie groups
- Shortest path between two points on a surface

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.

- Closed form for $n$th derivative of exponential of $f$
- Problem about limit of Lebesgue integral over a measurable set
- If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
- Is number rational?
- How to prove that the closed convex hull of a compact subset of a Banach space is compact?
- Alternate definition of Hilbert space operator norm
- Problems that become easier in a more general form
- Let $L$ be a Lie algebra. why if $L$ be supersolvable then $L'=$ is nilpotent.
- Homeomorphism of the Disk
- Some hints for “If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$”
- Can 12 teams in 6 disciplines play 6 rounds without repetition?
- Proof that every metric space is homeomorphic to a bounded metric space
- Seeking a combinatorial proof of the identity$1 f_1+2 f_2+\cdots+n f_n=n f_{n + 2} – f_{n + 3} + 2$
- Approximations for the number of divisors of an integer
- Is the closedness of the image of a Fredholm operator implied by the finiteness of the codimension of its image?