Geometries on Surfaces

by Burkard Polster , Günter Steinke

• ISBN13: 9780521660587
• ISBN10: 0521660580
• Format: Hardcover
• Copyright: 2002-01-07
• Publisher: Cambridge University Press

Summary

The projective, Möbius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces. This book summarizes all known major results and open problems related to these classical point-line geometries and their close (nonclassical) relatives. Topics covered include: classical geometries; methods for constructing nonclassical geometries; classifications and characterizations of geometries. This work is related to many other fields including interpolation theory, convexity, the theory of pseudoline arrangements, topology, the theory of Lie groups, and many more. The authors detail these connections, some of which are well-known, but many much less so. Acting both as a reference for experts and as an accessible introduction for graduate students, this book will interest anyone wishing to know more about point-line geometries and the way they interact.

Table of Contents

Preface	xvii
Geometries for Pedestrians	1	(22)
Geometries of Points and Lines	1	(8)
Projective Planes	2	(3)
Affine Planes	5	(1)
Benz Planes--the Original Circle Planes	6	(2)
Orthogonal Arrays	8	(1)
Geometries on Surfaces	9	(11)
(Ideal) Flat Linear Spaces	10	(3)
Flat Circle Planes	13	(2)
A Network of Relationships	15	(1)
Interpolation and the Axiom of Joining	16	(1)
Convexity	17	(1)
Topological Geometries on Surfaces	18	(1)
Classification with Respect to the Group Dimension	19	(1)
Definitions of Frequently Used Terms	20	(3)
Flat Linear Spaces	23	(114)
Models of the Classical Flat Projective Plane	24	(4)
The Euclidean Plane Plus Its Line at Infinity	25	(1)
The Geometry of Great Circles and the Disk Model	25	(2)
The Classical Point Mobius Strip Plane	27	(1)
Convexity Theory	28	(8)
Convex Curves, Arcs, and Ovals	31	(3)
Dependence of the Axioms of Joining and Intersection	34	(2)
Continuity of Geometric Operations and the Line Space	36	(8)
Isomorphisms, Automorphism Groups, and Polarities	44	(9)
Topological Planes and Flat Linear Spaces	53	(3)
Classification with Respect to the Group Dimension	56	(7)
Flat Projective Planes	57	(2)
R2-Planes	59	(2)
Mobius Strip Planes	61	(2)
Constructions	63	(40)
Original Moulton Planes	63	(2)
Semi-classical Planes and Generalized Moulton Planes	65	(1)
Radial Planes and Radial Moulton Planes	66	(1)
Shift Planes and Planar Functions	67	(3)
Arc Planes	70	(6)
Skew Hyperbolic Planes	76	(1)
Cartesian Planes	77	(1)
Strambach's SL2(R)-Plane	78	(1)
Integrated Foliations	79	(1)
Different Ways to Cut and Paste	80	(8)
Pasted Planes	88	(10)
Semioval Planes	98	(5)
The Modified Real Dual Cylinder Plane	103	(1)
Planes with Special Properties	103	(6)
Compact Groups of Automorphisms	103	(3)
More Rigid Planes	106	(1)
Differentiable Planes	106	(1)
Maximal Flat Stable Planes and the First Nonclassical Flat Linear Space	107	(2)
Other Invariants and Characterizations	109	(7)
The Lenz-Barlotti Types	110	(1)
Groups of Projectivities	111	(4)
Semigroups of Continuous Lineations	115	(1)
Related Geometries	116	(18)
Sharply Transitive Sets	116	(4)
Quasi-Sharply-2-Transitive Sets and Abstract Ovals	120	(5)
Semibiplanes	125	(4)
Pseudoline Arrangements, Universal Planes, Spreads	129	(5)
Open Problems	134	(3)
Spherical Circle Planes	137	(75)
Models of the Classical Flat Mobius Plane	137	(7)
The Geometry of Plane Sections	137	(1)
The Geometry of Euclidean Lines and Circles	138	(2)
Pentacyclic Coordinates	140	(1)
The Geometry of Chains	141	(2)
The Geometry of the Group of Fractional Linear Maps	143	(1)
Derived Planes and Topological Properties	144	(10)
Derived R2-Planes	145	(1)
Affine Parts	146	(1)
Continuity of Geometric Operations	146	(3)
Topological Mobius Planes	149	(2)
Circle Space and Flag Space	151	(3)
Constructions	154	(15)
Ovoidal Planes	154	(3)
Ewald's Planes	157	(2)
Semi-classical Flat Mobius Planes	159	(4)
Different Ways to Cut and Paste	163	(5)
Integrals of R2-Planes	168	(1)
Groups of Automorphisms and Groups of Projectivities	169	(16)
Automorphisms and Automorphism Groups	169	(6)
Compact Groups of Automorphisms	175	(3)
Classification with Respect to Group Dimension	178	(5)
Von Staudt's Point of View--Groups of Projectivities	183	(2)
The Hering Types	185	(10)
q-Translations	186	(2)
The Classification	188	(2)
Examples	190	(5)
Characterizations of the Classical Plane	195	(5)
The Locally Classical Plane	196	(1)
The Miquelian Plane	196	(2)
The Symmetric Plane	198	(1)
The Plane with Transitive Group	198	(1)
The Plane of Hering Type at Least V	199	(1)
Summary	200	(1)
Planes with Special Properties	200	(2)
Rigid Planes	201	(1)
Differentiable Planes	201	(1)
Subgeometries and Lie Geometries	202	(6)
Recycled Flat Projective Planes	202	(2)
Double Covers of R2-Planes and Flat Projective Planes	204	(2)
3-Ovals	206	(1)
Flocks and Resolutions	206	(2)
Open Problems	208	(4)
Toroidal Circle Planes	212	(77)
Models of the Classical Flat Minkowski Plane	213	(15)
The Geometry of Plane Sections	213	(1)
The Geometry of Euclidean Lines and Hyperbolas	214	(3)
The Pseudo-Euclidean Geometry	217	(2)
Pentacyclic Coordinates	219	(1)
The Geometry of the Group of Fractional Linear Maps	220	(2)
The Geometry of Chains	222	(4)
The Beck Model	226	(2)
Derived Planes and Topological Properties	228	(9)
Derived R2-Planes	229	(1)
Affine Parts	230	(1)
Standard Representation and Sharply 3-Transitive Sets	231	(1)
Continuity of the Geometric Operations	232	(2)
Topological Minkowski Planes	234	(3)
Circle Space and Flag Space	237	(1)
Constructions	237	(18)
The Two Halves of a Toroidal Circle Plane	238	(3)
Different Ways to Cut and Paste	241	(3)
Integrals of R2-Planes	244	(1)
The Generalized Hartmann Planes	244	(3)
The Artzy-Groh Planes	247	(4)
Modified Classical Planes	251	(3)
Proper Toroidal Circle Planes	254	(1)
Automorphism Groups and Groups of Projectivities	255	(13)
Automorphisms	256	(1)
Groups of Automorphisms	257	(1)
The Kernels	258	(3)
Planes Admitting 3-Dimensional Kernels	261	(1)
Classification with Respect to the Group Dimension	262	(4)
Von Staudt's Point of View--Groups of Projectivities	266	(2)
The Klein-Kroll Types	268	(7)
G-Translations	269	(2)
q-Translations	271	(1)
(p, q)-Homotheties	272	(2)
Some Examples	274	(1)
Characterizations of the Classical Plane	275	(6)
The Locally Classical Plane	275	(1)
The Miquelian Plane	275	(1)
The Plane with Many Desarguesian Derivations	276	(1)
The Plane in Which the Rectangle Configuration Closes	276	(1)
The Symmetric Plane	277	(2)
The Plane with Flag-Transitive Group	279	(1)
The Plane of Klein-Kroll Type at Least V, E, or 21	279	(1)
Summary	280	(1)
Planes with Special Properties	281	(2)
Rigid Planes	281	(1)
Differentiable Planes	282	(1)
Subgeometries and Lie Geometries	283	(4)
Flocks and Resolutions	283	(3)
Double Covers of Disk Mobius Strip Planes	286	(1)
Open Problems	287	(2)
Cylindrical Circle Planes	289	(71)
Models of the Classical Flat Laguerre Plane	290	(9)
The Geometry of Plane Sections	290	(1)
The Geometry of Euclidean Lines and Parabolas	290	(2)
The Geometry of Trigonometric Polynomials	292	(1)
The Geometry of Oriented Lines and Circles	292	(3)
Pentacyclic Coordinates	295	(1)
The Geometry of Chains	296	(3)
Derived Planes and Topological Properties	299	(5)
Derived R2-Planes	299	(1)
Affine Parts	300	(1)
Continuity of the Geometric Operations	301	(1)
Topological Laguerre Planes	302	(1)
Circle Space and Flag Space	303	(1)
Constructions	304	(25)
Ovoidal Planes	304	(2)
Semi-classical Flat Laguerre Planes	306	(6)
Different Ways to Cut and Paste	312	(4)
Integrals of Flat Linear Spaces	316	(8)
Planes of Generalized Shear Type	324	(1)
Planes of Translation Type	325	(1)
The Artzy-Groh Planes	326	(2)
Planes of Shift Type	328	(1)
Automorphism Groups and Groups of Projectivities	329	(13)
Automorphisms	329	(3)
The Kernel	332	(1)
Planes Admitting an at Least 3-Dimensional Kernel	333	(3)
Classification with Respect to the Group Dimension	336	(5)
Von Staudt's Point of View--Groups of Projectivities	341	(1)
The Kleinewillinghofer Types	342	(9)
C-Homologies	343	(1)
Laguerre Translations	344	(3)
(p, q)-Homotheties	347	(2)
Some Examples	349	(2)
Characterizations of the Classical Plane	351	(1)
Planes with Special Properties	352	(1)
Rigid Planes	352	(1)
Differentiable Planes	352	(1)
Subgeometries and Lie Geometries	353	(5)
Recycled Flat Projective Planes	353	(2)
Double Covers of R2-Planes and Flat Projective Planes	355	(1)
Flocks and Resolutions	356	(2)
Open Problems	358	(2)
Generalized Quadrangles	360	(35)
The Classical Antiregular 3-Dimensional Quadrangle	361	(3)
Basic Properties	364	(1)
From Circle Planes to Generalized Quadrangles and Back	365	(4)
Flat Laguerre Planes	365	(2)
Flat Mobius Planes	367	(2)
Flat Minkowski Planes	369	(2)
Sisters of Circle Planes	371	(4)
Sisters of Flat Laguerre Planes	372	(1)
Sisters of Flat Mobius Planes	373	(1)
Sisters of (Halves of) Flat Minkowski Planes	374	(1)
Flat Biaffine Planes and Flat Homology Semibiplanes	375	(12)
Flat Biaffine Planes in Flat Laguerre Planes	377	(3)
Flat Biaffine Planes in Flat Mobius Planes	380	(1)
Flat Homology Semibiplanes in Flat Laguerre Planes	381	(2)
Flat Homology Semibiplanes in Flat Mobius Planes	383	(2)
Split Semibiplanes	385	(2)
Different Ways to Cut and Paste	387	(1)
The Apollonius Problem	388	(7)
Tubular Circle Planes	395	(34)
Unisolvent Sets of Functions	396	(10)
Fibrated Circle Planes	397	(1)
Models of the Classical Tubular Circle Planes	398	(3)
Basic Properties	401	(5)
Nested (Ph)unisolvent Sets and Their Circle Planes	406	(14)
Unrestricted (Ph)unisolvent Sets	406	(2)
Integrating Unisolvent Sets	408	(1)
Nested Tubular Circle Planes	408	(8)
Automorphisms of Nested Tubular Circle Planes	416	(4)
Convexity and Cut-and-Paste Constructions	420	(7)
Convexity	421	(1)
Different Ways to Cut and Paste	422	(5)
Open Problems	427	(2)
Appendix 1 Tools and Techniques from Topology and Analysis	429	(15)
Appendix 2 Lie Transformation Groups	444	(14)
Bibliography	458	(25)
Index	483