Trivial vector bundle - Imported Knowlpedia Content

Let $M$ be a smooth manifold and let $V$ be a finite-dimensional real vector space.

The trivial vector bundle with fiber $V$ is the projection

\pi: M\times V \longrightarrow M,\qquad \pi(x,v)=x,

with fiberwise vector space operations

(x,v)+(x,w)=(x,v+w),\qquad \lambda(x,v)=(x,\lambda v).

It is a smooth vector bundle of rank $\dim V$ . A section of $M\times V$ is the same thing as a smooth map $s:M\to V$ .

The bundle admits a global frame: choosing a basis $(e_1,\dots,e_k)$ of $V$ yields sections $s_i(x)=(x,e_i)$ that span each fiber.

The trivial bundle is the vector-bundle analogue of the trivial principal bundle.

Examples

Euclidean tangent bundle. The tangent bundle of $\mathbb R^n$ is trivial: $T\mathbb R^n\cong \mathbb R^n\times \mathbb R^n$ using the standard coordinate vector fields.

Product bundles. For any manifold $M$ and any $k$ , the bundle $M\times\mathbb R^k\to M$ is the standard rank- $k$ trivial bundle; its sections are $k$ -tuples of smooth functions on $M$ .

Connections on trivial bundles. A connection on a vector bundle on $M\times V$ can be written globally in a chosen basis as $\nabla=d+A$ , where $A$ is a matrix of 1-forms on $M$ .