Nash function: Difference between revisions

Revision as of 22:42, 8 November 2007

Revision as of 13:32, 19 November 2007

Line 21:

that any compact [[differentiable manifold|''C''^∞ manifold]] admits a Nash manifold structure,

i.e., is diffeomorphic to some Nash manifold. More generally, a

''C''^∞∞ manifold admits a Nash manifold strucure if and only if it

is diffeomorphic to the interior of some compact ''C''^∞∞ manifold

possibly with boundary. Nash's result was later (1973) completed by Alberto

Tognoli who proved that any compact ''C''^∞∞ manifold is

diffeomorphic to some affine real algebraic manifold ; actually, any Nash

manifold is Nash diffeomorphic to an affine real algebraic manifold. These

results exemplify the fact that the Nash category is somewhat intermediate

between the ''C''^∞∞ and the algebraic categories.

==Local properties==

Line 45:

independently (1973) by Jean-Jacques Risler and Gustave Efroymson. Nash

manifolds have properties similar to but weaker than [[Cartan's theorems A and B]]

on [[Stein manifold|Stein manifolds]]s. Let <math>\mathcal{N}</math> denote the sheaf of Nash function germs on

a Nash manifold ''M'', and <math>\mathcal{I}</math> be a [[coherent sheaf]] of

<math>\mathcal{N}</math>-ideals. Assume <math>\mathcal{I}</math> is finite, i.e., there exists a

Line 63:

of the field of real numbers, and the above statements still hold. Abstract Nash functions

can also be defined on the real spectrum of any commutative ring.

== Sources ==

Line 73 ⟶ 72:

#J-J. Risler: Sur l'anneau des fonctions de Nash globales. C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1513--A1516.

#M. Shiota: Nash manifolds. Springer, 1987.

#A. Tognoli: Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 27 (1973), 167--185.

{{maths-stub}}

[[Category:geometry]]