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 ${\ displaystyle M}$ ${\ displaystyle x \ in M}$

{\ displaystyle \ exp _ {x}: T_ {x} M \ rightarrow M}

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: ${\ displaystyle M}$ ${\ displaystyle M}$

{\ displaystyle \ pi _ {j} (M) = 0 \ \ \ forall j \ geq 2}

.

Generalization (Metric Spaces)

Be a Hadamard room . Then there is a unique geodesic for all of them ${\ displaystyle X}$ ${\ displaystyle x, y \ in X}$

{\ displaystyle \ sigma _ {xy}: \ left [0,1 \ right] \ rightarrow X}

with , and constantly depends on and . ${\ displaystyle \ sigma _ {xy} (0) = x, \ sigma _ {xy} (1) = y}$ ${\ displaystyle \ sigma _ {xy}}$ ${\ displaystyle x}$ ${\ displaystyle y}$

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 ${\ displaystyle X}$ ${\ displaystyle {\ widetilde {X}}}$ ${\ displaystyle {\ tilde {d}}}$

the overlay is a local isometric drawing , and ${\ displaystyle {\ widetilde {X}} \ to X}$

{\ displaystyle ({\ widetilde {X}}, {\ tilde {d}})}

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

[1] Ballmann, op. Cit., Theorem I.4.5