Cartan–Hadamard theorem

From Wikipedia, the free encyclopedia

The Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Jacques Hadamard in 1898 for surfaces and generalized to arbitrary dimension by Élie Cartan in 1928 (see the first three references). The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; two detailed proofs were published in 1990 by Werner Ballmann for locally compact spaces and by Alexander–Bishop for general locally convex spaces.

Contents

[edit] Modern formulation

In modern metric geometry, the Cartan–Hadamard theorem is the statement that the universal cover of a connected non-positively curved complete metric space X is a Hadamard space. In particular, if X is simply-connected then it is a unique geodesic space, and hence contractible. The most general form of the theorem does not require the space to be locally compact, or even to be a geodesic space.

Let X be a complete metric space. Recall that X is called convex if for any two geodesics a(t) and b(t), the distance function

d: t \mapsto \operatorname{dist}(a(t), b(t))

is a convex function (in the metric geometry sense). A space is called locally convex if for every point p, there is an open neighbourhood of p which is a convex space with respect to the induced metric. If X is locally convex then it admits a universal cover.

[edit] Theorem

Let X be a connected complete metric space and suppose that X is locally convex. Then the universal cover of X is a convex geodesic space with respect to the induced length metric d. In particular, any two points of the universal cover are joined by a unique geodesic. Moreover, if X is a non-positively curved connected complete metric space, then its universal cover is CAT(0) with respect to d.

[edit] Significance

The Cartan–Hadamard theorem provides an example of a local-to-global correspondence in Riemannian and metric geometry: namely, a local condition (non-positive curvature) and a global condition (simply-connectedness) together imply a strong global property (contractibility); or in the Riemannian case, diffeomorphism with Rn.

The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is aspherical. This fact is of crucial importance for modern geometric group theory.

[edit] See also

[edit] References

  • Kobayashi, Shochichi; Nomizu, Katsumi (1969). Foundations of differential geometry, Vol. II, Tracts in Mathematics 15. New York: Wiley Interscience, xvi+470. ISBN 0-470-49648-7. 
  • Helgason, Sigurdur (1978). Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics 80. New York: Academic Press, xvi+628. ISBN 0-12-338460-5. 
  • Carmo, Manfredo Perdigão do (1992). Riemannian geometry, Mathematics: theory and applications. Boston: Birkhäuser, xvi+300. ISBN 0-8176-3490-8. 
  • Alexander, Stephanie B.; Bishop, Richard L. (1990). "The Hadamard-Cartan theorem in locally convex metric spaces". Enseign. Math. (2) 36 (3–4): 309–320. 
  • Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature, DMV Seminar 25. Basel: Birkhäuser Verlag, viii+112. ISBN 3-7643-5242-6.  MR1377265
  • Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319. Berlin: Springer-Verlag, xxii+643. ISBN 3-540-64324-9.  MR1744486