Dual vector bundle

Let $\pi:E\to M$ be a smooth vector bundle (real or complex) over a smooth manifold . The dual vector bundle of $E$ is the vector bundle

defined fiberwise by

with smooth structure characterized by the property that any local trivialization $E|_U\cong U\times \mathbb F^r$ induces a local trivialization

via fiberwise duality.

Equivalently, if $(e_1,\dots,e_r)$ is a local frame on $U$, then there is a uniquely determined dual local frame $(e^1,\dots,e^r)$ of $E^*|_U$ such that $e^i(e_j)=\delta^i_j$ pointwise; changes of frame are governed by inverse transpose (real case) or inverse conjugate transpose (Hermitian case).

A vector bundle morphism $\Phi:E\to F$ over $\mathrm{id}_M$ induces a dual morphism $\Phi^*:F^*\to E^*$ over $\mathrm{id}_M$ by precomposition on each fiber.

Examples

1. Cotangent bundle. The cotangent bundle $T^*M$ is the dual vector bundle of the tangent bundle $TM$.

2. Dual of a trivial bundle. $(M\times \mathbb F^r)^*\cong M\times (\mathbb F^r)^*$ canonically.

3. Dual line bundle. If $L\to M$ is a real or complex line bundle, then $L^*$ is again a line bundle; fiberwise, it consists of linear functionals on $L_x$. The tensor product $L\otimes L^*$ has a canonical nowhere-zero section given by evaluation, so it is canonically trivial.