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Orientation of a vector bundle

Generalization of an orientation of a vector space

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Source updated July 22, 2026

In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map)

\phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {R} ^{n}

is fiberwise orientation-preserving, where Rn is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant.

If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).

A vector bundle together with an orientation is called an oriented bundle. A vector bundle that can be given an orientation is called an orientable vector bundle.

The basic invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a Gysin sequence.

01Examples

A complex vector bundle is oriented in a canonical way.

The notion of an orientation of a vector bundle generalizes an orientation of a differentiable manifold: an orientation of a differentiable manifold is an orientation of its tangent bundle. In particular, a differentiable manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.)

02Operations

To give an orientation to a real vector bundle E of rank n is to give an orientation to the (real) determinant bundle \operatorname {det} E=\wedge ^{n}E of E. Similarly, to give an orientation to E is to give an orientation to the unit sphere bundle of E.

Just as a real vector bundle is classified by the real infinite Grassmannian, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces.

03Thom space

From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle E of rank n means a choice (and existence) of a class

u\in H^{n}(T(E);\Lambda )

in the cohomology ring of the Thom space T(E) such that u generates {\tilde {H}}^{*}(T(E);\Lambda ) as a free H^{*}(E;\Lambda )-module globally and locally: i.e.,

H^{*}(E;\Lambda )\to {\tilde {H}}^{*}(T(E);\Lambda ),x\mapsto x\smile u

is an isomorphism (called the Thom isomorphism), where "tilde" means reduced cohomology, that restricts to each isomorphism

H^{*}(\pi ^{-1}(U);\Lambda )\to {\tilde {H}}^{*}(T(E|_{U});\Lambda )

induced by the trivialization \pi ^{-1}(U)\simeq U\times \mathbf {R} ^{n}. One can show, with some work, that the usual notion of an orientation coincides with a Z-orientation.

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Sources and credits

This article is adapted from the Wikipedia article “Orientation of a vector bundle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Integration along fibers

In differential geometry, the integration along fibers of a k-form yields a {\displaystyle (k-m)} -form where m is the dimension of the fiber, via "integration". It is also called the fiber integration.

Orientation sheaf

In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is the local homology group o X , x = H n ⁡ {\displaystyle o_{X,x}=\operatorname {H} _{n}(X,X-\{x\})} (in the integer coefficients or some other coefficients). Let Ω M k {\displaystyle \Omega _{M}^{k}} be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf V M = Ω M n ⊗ o M {\displaystyle {\mathcal {V}}_{M}=\Omega _{M}^{n}\otimes {\mathcal {o}}_{M}} is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: ∫ M : Γ c ( M , V M ) → R .

Orientation sheaf

In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is the local homology group o X , x = H n ⁡ {\displaystyle o_{X,x}=\operatorname {H} _{n}(X,X-\{x\})} (in the integer coefficients or some other coefficients). Let Ω M k {\displaystyle \Omega _{M}^{k}} be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf V M = Ω M n ⊗ o M {\displaystyle {\mathcal {V}}_{M}=\Omega _{M}^{n}\otimes {\mathcal {o}}_{M}} is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: ∫ M : Γ c ( M , V M ) → R .