Torus
In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by , with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice).
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to be the ratio area/sys2. The systolic ratio SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry.
In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by , with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice).
A similar result is given by Pu's inequality for the real projective plane from 1952, due to Pao Ming Pu, with an upper bound of π/2 for the systolic ratio SR(RP2), also attained in the constant curvature case.
For the Klein bottle K, Bavard (1986) obtained an optimal upper bound of for the systolic ratio:
based on work by Blatter from the 1960s.
An orientable surface of genus 2 satisfies Loewner's bound , see (Katz-Sabourau '06). It is unknown whether or not every surface of positive genus satisfies Loewner's bound. It is conjectured that they all do. The answer is affirmative for genus 20 and above by (Katz-Sabourau '05).
For a closed surface of genus g, Hebda and Burago (1980) showed that the systolic ratio SR(g) is bounded above by the constant 2. Three years later, Mikhail Gromov found an upper bound for SR(g) given by a constant times
A similar lower bound (with a smaller constant) was obtained by Buser and Sarnak. Namely, they exhibited arithmetic hyperbolic Riemann surfaces with systole behaving as a constant times . Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times .
The study of the asymptotic behavior for large genus of the systole of hyperbolic surfaces reveals some interesting constants. Thus, Hurwitz surfaces defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound
resulting from an analysis of the Hurwitz quaternion order. A similar bound holds for more general arithmetic Fuchsian groups. This 2007 result by Mikhail Katz, Mary Schaps, and Uzi Vishne improves an inequality due to Peter Buser and Peter Sarnak in the case of arithmetic groups defined over , from 1994, which contained a nonzero additive constant. For the Hurwitz surfaces of principal congruence type, the systolic ratio SR(g) is asymptotic to
Using Katok's entropy inequality, the following asymptotic upper bound for SR(g) was found in (Katz-Sabourau 2005):
see also (Katz 2007), p. 85. Combining the two estimates, one obtains tight bounds for the asymptotic behavior of the systolic ratio of surfaces.
There is also a version of the inequality for metrics on the sphere, for the invariant L defined as the least length of a closed geodesic of the metric. In '80, Gromov conjectured a lower bound of for the ratio area/L2. A lower bound of 1/961 obtained by Croke in '88 has recently been improved by Nabutovsky, Rotman, and Sabourau.
In geometry, the circumference is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk. The circumference of a sphere is the circumference, or length, of any one of its great circles.
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960.
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.
In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it.
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality
In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983; it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.
In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve, is a compact Riemann surface of genus with the highest possible order of the conformal automorphism group in this genus, namely of order 48. The full automorphism group is the semi-direct product of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation
In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.
In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14. The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces.
Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length.
The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow. It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples.
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967 by Goro Shimura, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see Hurwitz quaternion.
In differential geometry, systolic freedom refers to the fact that closed Riemannian manifolds may have arbitrarily small volume regardless of their systolic invariants. That is, systolic invariants or products of systolic invariants do not in general provide universal lower bounds for the total volume of a closed Riemannian manifold.
Mikhail "Mischa" Gershevich Katz is an Israeli mathematician, a professor of mathematics at Bar-Ilan University. His main interests are differential geometry, geometric topology and mathematics education; he is the author of the book Systolic Geometry and Topology, which is mainly about systolic geometry. The Katz–Sabourau inequality is named after him and Stéphane Sabourau.
Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group . They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.
In mathematics, the Besicovitch inequality is a geometric inequality relating volume of a set and distances between certain subsets of its boundary. The inequality was first formulated by Abram Besicovitch.
