In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), 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 general linear group of matrices over the finite field ). The full automorphism group (including reflections) 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 . The Bolza surface is the smooth completion of the affine curve. Of all genus hyperbolic surfaces, the Bolza surface maximizes the length of the systole (Schmutz 1993). As a hyperelliptic Riemann surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six vertices of a regular octahedron inscribed in the sphere, as can be readily seen from the equation above.
The Bolza surface has attracted the attention of physicists, as it provides a relatively simple model for quantum chaos; in this context, it is usually referred to as the Hadamard–Gutzwiller model.[1] The spectral theory of the Laplace–Beltrami operator acting on functions on the Bolza surface is of interest to both mathematicians and physicists, since the surface is conjectured to maximize the first positive eigenvalue of the Laplacian among all compact, closed Riemann surfaces of genus with constant negative curvature.
^Aurich, R.; Sieber, M.; Steiner, F. (1 August 1988). "Quantum Chaos of the Hadamard–Gutzwiller Model". Physical Review Letters. 61 (5): 483–487. Bibcode:1988PhRvL..61..483A. doi:10.1103/PhysRevLett.61.483. PMID 10039347. S2CID 20390243.
mathematics, the Bolzasurface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 {\displaystyle...
4-fold rotational symmetry, Z4, [4]+. In Riemann surface theory, the Bolzasurface, sometimes called the Bolza curve, is obtained as the ramified double cover...
the full group of isometries of some Riemann surface. For genus 2 the order is maximized by the Bolzasurface, with order 48. For genus 3 the order is maximized...
non-orientable surface of genus two is the Klein bottle. The Bolzasurface is the most symmetric Riemann surface of genus 2, in the sense that it has the largest...
closed geodesics on M. Examples. the Bolzasurface of genus 2; the Klein quartic of genus 3; the Macbeath surface of genus 7; the First Hurwitz triplet...
Oskar Bolza (12 May 1857 – 5 July 1942) was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Palatinate, then a district...
surfaces of genus 14). More generally, it is the most symmetric surface of a given genus (being a Hurwitz surface); in this class, the Bolzasurface is...
surface: such straight lines are geodesics. One of the earliest cases studied is Hadamard's billiards, which describes geodesics on the Bolzasurface...
Integrale erster Gattung (in Mémoires des savants etc., vol. 11, 1851). Bolzasurface Superelliptic curve "Hyper-elliptic curve", Encyclopedia of Mathematics...
articles. Interesting examples are provided by the Bolzasurface, Klein quartic, Macbeath surface, First Hurwitz triplet. A family of optimal systolic...
Hurwitz surface (of genus 3) is the Klein quartic. The (2 3 8) triangle tiles the Bolzasurface, a highly symmetric (but not Hurwitz) surface of genus...
of the smooth completion, nor is it canonical. Hyperelliptic curve Bolzasurface Griffiths, 1972, p. 286. Conrad, Brian (2007). "Deligne's notes on Nagata...
John Wiley & Sons, Inc. 1998. (pp 130–133) ISBN 0-471-29198-6 Bolza 1967, p. 80 Bolza 1967, p. 88 Meggs, Philip B. (1998). A History of Graphic Design...
contact with a solid surface Early American publishers and printers The influence of the internet on the printing industry Bolza, Hans (1967). "Friedrich...
Giori established an immediate rapport with the chairman of Koenig, Hans Bolza, and while the deal they signed in 1952 was preliminary to a long-term agreement...
encouraged further development. In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques...
Göttingen: Dieterich. p. 9. Stäckel 1917, p. 110-119. Stäckel 1917, p. 105-106. Bolza 1921, p. 70-74. Bühler 1981, p. 100–102. Klein 1979, pp. 57–60. Schaaf 1964...
Interaction in the World Economy: The Roots of Modernity, Washington 2005, p.65 Bolza 1967; Gerhardt 1971; Gerhardt 1978, p. 217 The main source is Meyers Konversations-Lexikon...
discovered the fundamental properties of the asymptotic lines on the Kummer surface. They later investigated W-curves, curves invariant under a group of projective...
an elderly Arcona. Fred Tatasciore voices the Klatooinian gorg seller Bolza Grool and Dee Bradley Baker voices both Grevel, an irascible Aleena, and...
Solution Connected with the Surface of Revolution of Minimum Area in the calculus of variations, supervised by Oskar Bolza. She became the first woman...
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