1. See also

2. References

3. External links

In algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf o_X on X such that the stalk of o_X 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.

See also

• orientation of a manifold
• There is also a definition in terms of dualizing complex in Verdier duality; in particular, one can define a relative orientation sheaf using a relative dualizing complex.

References

• {{Citation | last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | author1-link=Masaki Kashiwara | title=Sheaves on Manifolds | isbn=3540518614 | year=2002 | publisher=Springer | location=Berlin}}

External links

• Two kinds of orientability/orientation for a differentiable manifold

• He For She
• H. E. Freund
• Hefsek taharah
• Hefta Lake
• Hefty (Hefty Records)
• Hefty Stuart
• Heftzi-Ba
• H. E. Fuller
• Hefzabad, North Khorasan
• Hegan
• Hegang, Heilongjiang
• Hegar
• He Garden
• Hegar dilator
• Hegarty
• He-gassen
• Hegau-Ablachtal Railway
• Hegau-Gymnasium
• Hegazi
• Hegazy
• Hegdoli
• Hegdorn
• Hege Boekko
• Hegedüs Csilla
• Hege Hansen