Exterior power bundle

The vector bundle whose fiber at each point is the k-th exterior power of the original fiber.

Exterior power bundle

Let $\pi:E\to M$ be a smooth vector bundle of rank $r$ over a smooth manifold . For an integer $k$ with $0\le k\le r$ , the k-th exterior power bundle of $E$ is the vector bundle

\Lambda^k E \to M

defined fiberwise by

(\Lambda^k E)_x := \Lambda^k(E_x).

If $(e_1,\dots,e_r)$ is a local frame of $E|_U$ , then the wedge products

e_{i_1}\wedge \cdots \wedge e_{i_k}\qquad (1\le i_1<\cdots<i_k\le r)

form a local frame of $(\Lambda^k E)|_U$ . Under a change of local frame with transition matrix $g:U\cap V\to \mathrm{GL}(r,\mathbb F)$ , the induced transition matrix on $\Lambda^kE$ is the $k$ -th exterior power representation $\Lambda^k g$ .

The construction is functorial: a vector bundle morphism $\Phi:E\to F$ over $\mathrm{id}_M$ induces $\Lambda^k\Phi:\Lambda^kE\to \Lambda^kF$ fiberwise.

This exterior-power construction is compatible with the tensor product bundle viewpoint via universal properties.

Examples

Forms as sections. Taking $E=T^*M$ , the sections of $\Lambda^k T^*M$ are precisely smooth differential k-forms on $M$ .
Determinant line bundle. For a rank $r$ bundle $E$ , the top exterior power $\Lambda^r E$ is a line bundle, often denoted $\det(E)$ . An orientation of a real bundle can be encoded as a choice of “positive” component in $\det(E)\setminus\{0\}$ .
Low-degree cases. $\Lambda^0E$ is canonically the trivial line bundle $M\times \mathbb F$ , and $\Lambda^1E\cong E$ canonically.