Hadamard room

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 full metric space . ${\displaystyle X}$

By definition, a Hadamard space is exactly if it is a CAT (0) space, that is, if it is a geodetic metric space and all geodetic triangles are at least as thin as their comparison triangles in the Euclidean plane . The latter condition can be reformulated into the condition ${\displaystyle X}$

${\displaystyle d^{2}(z,m)\leq {\frac {1}{2}}(d^{2}(z,x)+d^{2}(z,y))-{\frac {1}{4}}d^{2}(x,y)}$

for all , where the center of the geodesic is between and . ${\displaystyle x,y,z\in X}$ ${\displaystyle m\in X}$ ${\displaystyle x}$ ${\displaystyle y}$

The following equivalent definition goes back to Bruhat-Tits:

A complete metric space is a Hadamard space if and only if there is a “center point” for each pair of points such that ${\displaystyle X}$ ${\displaystyle x,y\in X}$ ${\displaystyle m\in X}$

${\displaystyle d^{2}(z,m)\leq {\frac {1}{2}}(d^{2}(z,x)+d^{2}(z,y))-{\frac {1}{4}}d^{2}(x,y)}$

applies to all . ${\displaystyle z\in X}$

Examples

• Hadamard manifolds : simply connected, complete Riemannian manifolds of non-positive section curvature
• metric trees
• Bruhat Tits Building
• Hilbert rooms

properties

A generalization of the Cartan-Hadamard theorem applies to Hadamard spaces . For any one there is a unique geodesic with . The geodesic depends continuously on and . ${\displaystyle x,y\in X}$ ${\displaystyle \sigma _{xy}:\left[0,1\right]\rightarrow X}$ ${\displaystyle \sigma _{xy}(0)=x,\sigma _{xy}(1)=y}$ ${\displaystyle \sigma _{xy}}$ ${\displaystyle x}$ ${\displaystyle y}$

Furthermore, all properties of CAT (0) spaces apply to Hadamard spaces .

literature

• Werner Ballmann : Lectures on spaces of nonpositive curvature (= DMV seminar. 25). With an appendix by Misha Brin. Birkhäuser, Basel et al. 1995, ISBN 3-7643-5242-6 ( online (PDF; 818 kB) ).
• Sergei Buyalo, Viktor Schroeder: Spaces of Curvature Bounded Above. In: Jeffrey Cheeger, Karsten Grove (Ed.): Metric and Comparison Geometry (= Surveys in Differential Geometry. 11). International Press, Sommerville MA 2007, ISBN 978-1-57146-117-9 , pp. 295–328, doi : 10.4310 / SDG.2006.v11.n1.a10 .
• François Bruhat , Jacques Tits : Groupes reductifs sur un corps local. Chapitre 1. Données radicielles valuées. In: Publications Mathématiques de l'IHES. Vol. 41, 1972, ISSN  0073-8301 , pp. 5-251, doi : 10.1007 / BF02715544 .

Web links

• Jacob Lurie : Notes on the Theory of Hadamard Spaces (PDF; 308 kB)