Cartan-Hadamard's theorem
In mathematics , the Cartan-Hadamard theorem is a theorem of Riemannian geometry that describes the topology of manifolds of non-positive intersectional curvature . The statement is named after the mathematicians Élie Cartan and Jacques Hadamard . Hadamard proved it in 1898 for surfaces , and Cartan in 1928 for Riemannian manifolds in general .
statement
Let be a complete Riemann manifold of non-positive section curvature , then for each is the exponential map
an overlay .
Corollary : Let be a complete Riemannian manifold of non-positive cutting curvature, then is aspheric , i.e. H. the higher homotopy groups disappear:
- .
Generalization (Metric Spaces)
Be a Hadamard room . Then there is a unique geodesic for all of them
with , and constantly depends on and .
Local CAT (0) spaces
A complete, connected, metric space is locally called CAT (0) if every point has a neighborhood which (with the restricted metric) is a CAT (0) space .
A generalization of the Cartan-Hadamard theorem states that if there is a local CAT (0) space, then there is a unique metric on the universal overlay such that
- the overlay is a local isometric drawing , and
- is a CAT (0) space.
literature
- Werner Ballmann : Lectures on spaces of nonpositive curvature. (PDF; 818 kB) With an appendix by Misha Brin. DMV Seminar, 25th Birkhäuser Verlag, Basel, 1995. ISBN 3-7643-5242-6
- Manfredo Perdigão do Carmo : Riemannian geometry. Mathematics: theory and applications, Boston: Birkhäuser, 1992. ISBN 0-8176-3490-8
Individual evidence
- ↑ Ballmann, op. Cit., Theorem I.4.5