Soul set

The soul theorem is a mathematical theorem from the field of differential geometry , which enables the investigation of (non-compact) manifolds of nonnegative curvature to be traced back to the investigation of vector bundles over compact manifolds of nonnegative curvature. It was proven by Jeff Cheeger and Detlef Gromoll and, in its tightened form (for at least one point positive curvature) by Grigori Perelman .

Soul set

${\displaystyle M}$be a complete connected Riemannian manifold of nonnegative sectional curvature . Then there is a compact , totalkonvexe , totalgeodätische submanifold so that the normal bundle of is diffeomorphic. is called the soul of . ${\displaystyle S}$ ${\displaystyle M}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle M}$

If non-compact and the cutting curvature is strictly positive in at least one point, then the soul is a point, i.e. diffeomorphic to . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$

literature

• Cheeger, Gromoll: On the structure of complete manifolds of nonnegative curvature , Annals of Mathematics 96, 413-443, 1972.
• Perelman: Proof of the soul conjecture of Cheeger and Gromoll , Journal of Differential Geometry 40, 209-212, 1994.