Differential 1-form

From Diffgeom
Revision as of 19:38, 18 May 2008 by Vipul (talk | contribs) (2 revisions)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds

Definition

Symbol-free definition

A differential 1-form on a differential manifold is defined in the following ways:

  • It is a section of its cotangent bundle
  • It associates in a smooth manner, a linear functional on the tangent space at every point

We can also talk of a differential 1-form restricted to an open subset, which is a section of the restriction of the cotangent bundle to the open subset. This gives rise to the notion of the sheaf of differential 1-forms

Definition with symbols

Let M be a differential manifold. A differential 1-form on M is defined in the following equivalent ways:

  • It is an element of \Gamma(T^*M). Here T^*M denotes the cotangent bundle of M
  • It associates, to every point p \in M, a linear functional on T_p(M), in a smooth manner
  • it is a smooth map from TM to \R, such that the restriction to any fiber T_p(M), is a linear map.

Related notions

There is a related notion of a vector space-valued differential 1-form. This is the analog of differential 1-form, taking values in a vector space, rather than in real numbers.