Trivial vector bundle

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

with fiberwise vector space operations

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

1. 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.

2. 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 $M$.

3. 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 $M$.