Finiteness theorem: Difference between revisions

Revision as of 11:29, 23 June 2007

This article describes a result related to the sectional curvature of a Riemannian manifold

Statement

Let $n\in \mathbb {N}$ , $\Lambda <\infty$ , $D<\infty$ m and $v>0$ . Then consider the space $S$ of differential manifolds $M$ admitting a metric $g$ for which:

$dim(M)=n,|K|\leq \Lambda ,diam(M)\leq D,vol(M)\geq v$

where $K$ denotes the sectional curvature.

Then, $S$ consists of finitely many diffeomorphism classes/types.

@@ Line 5: / Line 5: @@
 Let <math>n \in \mathbb{N}</math>, <math>\Lambda < \infty</math>, <math>D < \infty</math>m and <math>v > 0</math>. Then consider the space <math>S</math> of differential manifolds <math>M</math> admitting a metric <math>g</math> for which:
-<math>dum(M) = n, |K| \le \Lambda, diam(M) \le D, vol(M) \ge v</math>
+<math>dim(M) = n, |K| \le \Lambda, diam(M) \le D, vol(M) \ge v</math>
 where <math>K</math> denotes the [[sectional curvature]].
 Then, <math>S</math> consists of finitely many diffeomorphism classes/types.