Orientation Sheaf

(Orientation (Geometry)

Tech

sciences
Orientation (Geometry
Link to Dbpedia

What is Orientation sheaf?

In 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 (in the integer coefficients or some other coefficients). Let be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf 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: If M is oriented; i.e., the orientation sheaf of the tangent bundle of M is literally trivial, then the above reduces to the usual integration of a differential form.

Technology Types

algebraic topologyorientation (geometry

Synonyms

Orientation bundle

Tech Info


Source: [object Object]
 — Date merged: 11/6/2021, 1:32:49 PM
 — Date scraped: 5/20/2021, 6:05:11 PM