Systole (math)

Shortest non-contractible curve on a torus.

In mathematics , the systole is an invariant of metric spaces .

definition

Be a compact metric space . Then the systole of is defined as the length of a shortest non-contractible closed curve in . ${\displaystyle X}$ ${\displaystyle \operatorname {sys} (X)}$ ${\displaystyle X}$ ${\displaystyle X}$

Here, a closed curve is a continuous mapping with . It is called contractible if there is a point and a homotopy with and for all . Otherwise it is called non-contractible. In a compact metric space, a shortest non-contractible curve is always a closed geodesic . Is simply connected , then every closed curve is contractible. In this case is for each metric . ${\displaystyle \gamma \colon \left[0,1\right]\rightarrow X}$ ${\displaystyle \gamma (0)=\gamma (1)}$ ${\displaystyle x\in X}$ ${\displaystyle H\colon \left[0,1\right]\times \left[0,1\right]\rightarrow X}$ ${\displaystyle H(t,0)=\gamma (t)}$ ${\displaystyle H(t,1)=x}$ ${\displaystyle t\in \left[0,1\right]}$ ${\displaystyle X}$ ${\displaystyle \operatorname {sys} (X)=\infty }$

Pu's inequality

The following applies to every Riemannian metric on the projective level ${\displaystyle g}$ ${\displaystyle \mathbb {R} P^{2}}$

${\displaystyle sys(\mathbb {R} P^{2},g)\leq {\frac {\pi }{2}}\operatorname {area} (g)}$,

where denotes the area and the systole of the metric. ${\displaystyle \operatorname {area} (\mathbb {R} P^{2},g)}$ ${\displaystyle \operatorname {sys} (\mathbb {R} P^{2},g)}$

Loewner's inequality

The inequality applies to every Riemannian metric on the 2-dimensional torus ${\displaystyle g}$ ${\displaystyle T^{2}}$

${\displaystyle \operatorname {sys} (T^{2},g)\leq {\sqrt {{\frac {2}{\sqrt {3}}}\operatorname {area} (T^{2},g)}}}$,

where denotes the area and the systole of the metric. ${\displaystyle \operatorname {area} (T^{2},g)}$ ${\displaystyle \operatorname {sys} (T^{2},g)}$

Gromov's inequality

There is one universal constant that only depends on , so that for every aspherical -dimensional Riemannian manifold the inequality ${\displaystyle n}$ ${\displaystyle C_{n}}$ ${\displaystyle n}$ ${\displaystyle (M,g)}$

${\displaystyle \operatorname {sys} (M,g)\leq (C_{n}\operatorname {Vol} (M,g))^{\frac {1}{n}}}$

applies.

In particular, one has the inequality for areas

${\displaystyle sys(S,g)\leq {\frac {\pi }{2}}area(S,g)}$

with equality only for surfaces of constant curvature. For the torus this result improves Löwner's inequality.

Gromow's inequality holds more generally for essential manifolds ; H. if the classifying map induces a nontrivial homomorphism . ${\displaystyle M\rightarrow B\pi _{1}M}$ ${\displaystyle H_{n}(M,\mathbb {Z} /2Z)\rightarrow H_{n}(B\pi _{1}M,\mathbb {Z} /2\mathbb {Z} )}$

Web links

• Larry Guth : Metaphors in systolic geometry (PDF; 558 kB)
• Marcel Berger : What is ... a systole? , Notices of the AMS 55, 2008

literature

• Marcel Berger : Systoles et applications selon Gromov. Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. 771, 5, 279-310.
• Michail Leonidowitsch Gromow : Systoles and intersystolic inequalities. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291-362, Sémin. Congr., 1, Soc. Math. France, Paris, 1996.
• Gromow: Metric structures for Riemannian and non-Riemannian spaces. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhauser Boston, Inc., Boston, MA, 1999. ISBN 0-8176-3898-9
• Berger: A panoramic view of Riemannian geometry. Springer-Verlag, Berlin, 2003. ISBN 3-540-65317-1
• Michail G. Katz: Systolic geometry and topology. With an appendix by Jake P. Solomon. Mathematical Surveys and Monographs, 137. American Mathematical Society, Providence, RI, 2007. ISBN 978-0-8218-4177-8