Hadamard space

A Hadamard space is a mathematical object from the geometry of metric spaces. It is named after the mathematician Jacques Hadamard.

Definition

A Hadamard space is a complete CAT (0 ) space.

Equivalent definitions

Be a complete metric space.

By definition, a Hadamard space if and only if it is a CAT ( 0) - space, ie if it is a geodesic metric space and all geodesic triangles are at least as thin as their comparison triangles in the Euclidean plane. The latter condition can be rephrased in the condition

For all, with the midpoint of the geodesic between and inscribed.

On Bruhat - Tits following equivalent definition is derived:

Applies to all.

Examples

  • Hadamard manifolds: simply connected, complete Riemannian manifolds of positive sectional curvature not
  • Metric trees
  • Bruhat - Tits building
  • Cayley graphs of hyperbolic groups and general CAT ( 0 ) groups
  • Hilbert spaces

Properties

For Hadamard spaces, a generalization of the theorem of Cartan -Hadamard applies. Continue to apply for Hadamard spaces all properties of CAT ( 0) - spaces.

Hans Werner Ballmann Jacques Tits Jacob Lurie
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