Sheaf of connection algebras: Difference between revisions

Revision as of 20:53, 6 April 2008

This article defines a sheaf that can be associated to a differential manifold. The global analog of this sheaf, which is also the same as the object of the sheaf associated to the whole manifold, is: connection algebra

Definition

Let $M$ be a differential manifold. The sheaf of connection algebras of $M$ is defined as follows:

For every open subset $U$ of $M$ , the object associated to $U$ is the connection algebra associated to $U$ , viewed as a manifold.
The restriction map is defined using the natural restriction map for the sheaf of first-order differential operators, and the sheaf of infinitely differentiable functions

Sometimes the sheaf of connection algebras is termed the connection algebra, though the latter term is sometimes used for the global object.

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 ==Definition==
-Let <math>M</math> be a [[differential manifold]] and <math>E</math> a [[vector bundle]] over <math>M</math>. The '''sheaf of connection algebras''' of <math>E</math> is defined as follows:
+Let <math>M</math> be a [[differential manifold]]. The '''sheaf of connection algebras''' of <math>M</math> is defined as follows:
-* For every open subset <math>U</math> of <math>M</math>, the object associated to <math>E</math> is the [[connection algebra]] associated to the restriction of <math>E</math> to a vector bundle over <math>U</math>
+* For every open subset <math>U</math> of <math>M</math>, the object associated to <math>U</math> is the [[connection algebra]] associated to <math>U</math>, viewed as a manifold.
-* The restriction map is defined as follows: {{fillin}}
+* The restriction map is defined using the natural restriction map for the [[sheaf of first-order differential operators]], and the [[sheaf of infinitely differentiable functions]]
 Sometimes the '''sheaf of connection algebras''' is termed the connection algebra, though the latter term is sometimes used for the ''global object''.