Geometry and Differential Geometry: Proceedings of a Conference Held at the University of Haifa, Israel, March 18–23, 1979 PDF
Preview Geometry and Differential Geometry: Proceedings of a Conference Held at the University of Haifa, Israel, March 18–23, 1979
Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 792 yrtemoeG dna laitnereffiD yrtemoeG Proceedings of a Conference Held at the University of Haifa, Israel, March 18-23, 1979 detidE yb .R Artzy and .I Vaisman galreV-regnirpS Berlin Heidelberg New York 1980 Editors Rafael Artzy Izu Vaisman Department of Mathematics University of Haifa 31999 Haifa Israel AMS Subject Classifications (1980): 51-XX, 52-XX, 53-XX, 57-RX ISBN 3-540-09976-X Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09976-X Springer-Verlag NewYork Heidelberg Berlin This work the whole whether or is subject All to copyright. rights are reserved, part of specifically those is concerned, the material of reprinting, translation, re-use of reproduction broadcasting, photocopying by illustrations, machine or similar means, dna storage ni data § Under 54 banks. of Copyright the German waL copies where made era for fee a use, than private other is to the payable to fee the of amount the publisher, eb the publisher. with agreement by determined © yb Berlin Springer-Verlag Heidelberg 1980 Printed ni Germany Printing dna binding: Offsetdruck, Beltz Hemsbach/Bergstr. 0413/1412 012345- ECAFERP ,yrtemoeG under its various aspects, sah neeb a fascinating intellectual activity during the whole history of civillsed mankind. It is na old dna always wen science which, undoubtedly, sah pro- vided su wlth na important part of our about knowledge the world dna with ynam applications. The present volume includes the,text of most of the lectures presented at a Conference on Geometry and Differential Geometry, which was held at the University of Haifa, Israel, on March 18-23, 1979. The Conference was attended by some 70 mathematicians from all over the world, The subject matters covered a broad range, and many of the aspects of modern research in the fleld were discussed. This is why we decided to publish the Proceedings of the Conference and it is our hope that they will be of interest to the mathe- matical community. ehT organizers of the are Conference glad to acknowledge here their gratitude to the participants in the Conference, to all those ohw contributed to its success dna to Springer-Verlag ohw kindly accepted this volume In its Lecture Notes Series ft. Artzy, I. Valsman ELBAT F OSTNETNOC .I Geometry .R Artzy, On free Minkowski planes .F Bachmann, Rigidity ni the geometry of involutory elements of a group .W Benz, Ein Trennungsaxiom ni der Orthogonalgeometrie und eine Charakterisierung der reellen Ebene 41 L.Ja. Beresina, Applications of the theory of surfaces to the theory of graphs 2O .F Bonetti and .G Lunardon, Central translation S-spaces 24 oW Burau, Systems of quadrics through a general variety of Segre and their reduction to irreducible parts 3O E.W. Ellers, Generators and relations for classical groups 40 .P Erd~s, Some combinational problems ni geometry 46 .G Ewald, Uber die algorithmische L~sung des Steinitzproblems einer inneren Kennzeichnung polytopaler Sph~ren 54 H.-R. Halder, Regular permutation geometries 59 .A Herzer, On characterisations of kinematic spaces by parallelisms 16 .Y flamed, On realizations of vector products by polynomials which are identities for matrix rings 86 .J Joussen, On the construction of archimedean orders of a free plane 37 .H Karzel, Rectangular spaces 97 .H K~hlbrandt, On sharply 2-transitive permutation sets 29 .J Misfeld and .H Tecklenburg, Dimension of nearaffine spaces 79 oG Nicoletti, Generating cryptomorphic axiomatizations of matroids 011 .G Pickert, Partial planes with exactly two complete parallel classes t14 °G Pickert, A problem of free mobility 821 H.J. Samaga, A unified approach to Miquel's theorem and sti degenerations 132 .H Schaeffer, Automorphisms of Laguerre geometry and cone-preserving mappings of metric vector spaces 341 IV .R Schrarnm, Bounds for the number of solutions of certain piecewise linear equations 148 .W Seier, Zur Translationstransitlvit~t ni affinen Hjelmslevebenen 167 M.J. Thomsen, Near-rings with right inverse property 174 .H Zeitler, On reflections ni Minkowski planes 183 .II Differential Geometry D.E. Blair, On the space of Riemannian metrics on surfaces and contact manifolds 203 .R Blum, Circles on surfaces ni the Euclidean 3-space 213 .A Crumeyrolle, Classes caract~ristiques principales et secondaires 222 .T Duchamp and .M Kalka, Deformation theory and stability for holomorphic foliations 235 .J Girbau, Vanishing theorems and stability of complex analytic foliations 247 .A G~ay and .L Vanhecke, Power series expansions, differential geometry of geodesic spheres and tubes, and mean-value theorems 252 .Z Har'El, On distance-decreasing collineations 260 .H Kitahara, On a parametrix form ni a certain V-submersion 264 .W Klingenberg, Stable and unstable motions on surfaces 299 .Y Kosmann-Schwarzbach: Vector fields and generalized vector fields on fibered manifolds 307 .P Lecomte, Lie algebras of order 0 on a manifold 356 .P Libermann, Introduction ~ l'~tude de certains syst~mes dlffer~ntiels 363 V.I. Oliker, Infinitesimal deformations preserving parallel normal vector fields 383 J.F. Pornmaret, Differential Galols theory 406 G.M. Rassias, Counterexamples to a conjecture of Ren~ nrohT 414 .F Tricerri and .L Vanhecke, Conformal invariants on almost Hermitian manifolds 422 .I Valsman, Conformal changes of almost contact metric structures 435 ON FREE MINKOWSKI PLANES R. Artzy University of Haifa 31999 Haifa, Israel The structure of the automorphism groups of free planes has been determined for various geometries; however, this was successfully done only in cases of minimal generation, that is, when the plane originates by free generation from a minimal fundamental configura- tion. The results so far have been: Plane Fundamental configuration Automorpbism group Reference Projective ~ independent points <$4"$4;D#> [5] Affine 3 independent points <D3*D4;S2> [1] M~bius I circle c, 5 points 2 $2*¥4 of which lie on c [~] Laguerre I circle c, 3 mutually V~'V4"V 4 nonparallel points 2 of (contains an which lie on c error: one of these products is indeed not free) Here Sm, Dn, 4 V are, respectively, the symmetric group on m objects, the dihedral group of order 2n, and Klein's #-group. By <A.B; C> we mean the free product of A and B with C amalgamated. In the M6bius and Laguerre planes, as also in the Minkowski plane in the following, circles do not necessarily intersect (in terms of [6], the circle structures are "affine"). The procedures used in our approach to finding the structure of the automorphism group of the minimally generated free Minkowski plane will be based on previous extensive work by various authors: The theory of free extensions in general was thoroughly investigated by Schleiermacher and Strambach [6], and the case of the Minkowski plane was worked out by Heise and SSrensen [23. The practice for determining the automorphism groups was first developed by Sandler [5] and later refined by Iden [3,4]. We will follow these sources, their definitions and terminology rather closely. A few definitions follow: The circle containing the points A, B, C is called ABC . Parallelism of points is denoted by II + and II _. A couple (P,c) of a point P and a circle c incident with it is called a flag. The set of all points II to a given point is called a gene- + rator line, and so is the set of all points II to the point. Two circles c and d are called tangent or touching if IcNdl=1. We now start out from a minimal fundamental configuration consisting of 3 mutually nonparallel points B11 , B22 , B33 in a Minkowski plane. We perform the following operations ("steps") successively: I. Draw all lines through triples of mutually nonparallel existing points, II. draw all possible tangents to flags (P,c) already obtained through already obtained points Q such that P~ Q ~ c, III. intersect obtained circles with obtained +generator and -generator lines, IV. intersect intersecting but nontangent circles once more, V. draw all genrator lines through given points, VI. intersect all +generator lines with all -generator lines. Each full set of these 6 steps will be called a "stage". We start with stage O: Steps I through VI operating on the fundamental configuration yield 6 points Bi~ in addition to B11 , B22, B35: let BjjIl+BijlI_Bii; I,j=I,2,5. Thus, at the end of stage 0 we have 9 points, that is, 36 ordered point triples which can serve as fundamental triples. Definition. A point triple which can serve as fundamental triple (that is, which yields by free extension the same set of points as B11 , B22 , B33) will be called a generating triple (in short, GT). GT's will be designated by brackets. Proposition I. Stage 0 provides 36 ordered GT's. Now we begin stage I. Step I produces 6 circles BIiB2jB3k, where (ijk) is a permutation of (123). By step II we obtain 2 tangents to each of the 3 flags on each circle, that is, 6 distinct tangents to each of the 6 circles, a total of 56 such tangents. Each of these tangents intersects, by step III, two of the generator lines obtained so far. Thus we obtain 72 new points in stage I. Pr~Rosition 2. After completion of step III, one has in stage 4, 72.6=432 ordered GT's. Proof. Step III yields 2 points on each tangent in addition to the two points used in step II. Thus we have on each tangent 4 triples of mutually nonparallel points. However, such a triple is a GT if and only if two of its points are of the type BiN. Hence each of the 36 tangents produces just two GT's, a total of 72 GT's, or 72.6 ordered GT's. By inspection one sees also that no other combination of points produces GT's. Step IV requires that two intersecting circles, say d and b, can be shown not to be tangent. This is possible if it can be proved that at their point of intersection, say P, another circle, say c, touches b and intersects d at Q I P. Then d is not the tangent to b through Q at P, and hence d and b must intersect once more, say at R. The first figure describes this situation for the case where P is the point at infinity in the hyperbola model of the Minkowski plane. b / c (> 32B ~ 2 3 B 22B Proposition 3. By step IV, 18 GT's result from eackGT. Proof. Let B11 play the role of P, and for the sake of illustration choose P again as the point of infinity in the hyperbola model. Then we get the second figure. For P=B11 we obtain the 5 marked new points. Of these, only (BIIB23B32~BIIB22B33)~ B11 := M can be used for GT's. There are indeed two GT's involving M, namely [B53B22M] and [B52B23M]. For every choice of P among the nine Bij , we would obtain again one new point and two GT's, a total of 18 GT's. Proposition 4.Stagelprovides a total of 36.93 ordered GT's. Proof. In view of Proposition 3, we get from the GT's of Proposition 2, 72.18 GT's. By steps V and VI each pair of triples of the type [B22B33M] and [B23B52M] yields 6 GT's. Thus we have, respectively, 18.3 and 72.18.3 GT's, together (54+72.54)6 ordered GT's. From Proposition 2, we have 432 ordered GT's, which after steps V and VI make 432.6 ordered GT's. The total is (432+54+72.54)6=36.93 ordered GT's from stage 1. Remark. A consequence of Proposition 4 is that the number of ordered GT's of stage k > 2 is that of stage k-1 multiplied by 93, minus the = number of the GT's where (i) step IV is performed twice in a row on the same point triple and therefore cancels itself out, (ii) a tangent is drawn to a circle in stage k-l, and in stage k the original circle is regained as a tangent to the tangent. We now introduce collineations induced by permutations of the Bij. Define r, s, u, v ~s collineations induced, respectively, by the permut~ions as follows: [B11B22B33] maps under r on [B11B23B32], under s on [B15B21B32] , under u on [B11B32B23] , and under v on [B}dBf12B23]. By abuse of language we may write, in cycle notation r=(B22B23)(B53B32)(B13B12 ) s=(B31B33B32)(B21B25B22)(B1flB15B12) u=(B22B32)(B33B23)(B21B31 ) v=(B31B21Bdl)(B33B25B15)(B32B22Bq2)" We also define I t as the collineation which maps Bll on itself, B33 on B23 , and B22 on the point C such that CII+B22 and such that C lies on the circle c through B23 and tangent to B11B22B}} at Bll. Similarly, 2 t is defined as mapping B11 on itself, B22 on B2} , and B}} on the point D such that DII_B}} and such th~ D lies on the same tangent c. Finally, we remark that in a free plane C # B}2 and that, therefore, the circle B11B}2B25 is not the tangent to B11B22B}} at B11 through B25. Hence BllB22B}} and Bq1B32B25 have to intersect in a point M distinct from Bll. Again, we define a collineation m such that it maps [Bf11B22B}}] on [MB22B}5]. Proposition 5. For the collineations defined above, 2 2 } = = = r u s ~v 5 = (rs)2 = (uv) 2 = (rutfl)2 = (rut2)2 2 = = m (mr) 2 = (mu) 2 = 1. Moreover, r and s commute with u and v. Proof. The first 4 relations are immediate, in view of the cycle lengths of the permutations. We get rs=(B11B1})(B}}B}I)(B2}B21) and uv=(BIIB}I)(B}}B1})(B12B}2), hence they are involutory. We have B12B}})=vr' su=(B11B}}B22 () 32 21 15 () 31 23 12 = = 11 15 12 (B22B}1B2}B}2B21B}})=vs. Under rut1, [B11B22B}}] maps on [BIIB2}C]; when rutlis applied again, B11 maps on itself, and B25 on B22. Under ru, C maps on a point Eli+ B}} which lles on the tangent to Bd1B22B}} at Bll through B}2. Finally, under tl, E goes into the point If+B2} which lles on the tangent to B11CB23 at Bll through B22. This, however, is the point B33 , which proves that (rutl)2=1. Similarly, rut 2 takes [B11B22B33] into [Bq1DB25], and [B11DB23]rut2=[B1qB22B33], implying (rut2)2=1. By its very definition, m maps M on Bli , and hence m2=1. We have [BIIB22B33](mr)2=[MB23B32]mr=KB11B22B35] and [B11B22B33S(mu)2 = =[MB32B23]mu=[BIIB22B33] , hence (mr)2=(mu)2=l. Proposition 6. Every word composed of r, s, u, v, tq, t2, m is of the form n-1 ( ~=~ pn_j me..j 6n_jt~:~ ~ )Po with ~k 6 ~r,s,u,v~, 6k~ ~s,v~, ik~ ~,2~, ~k and ek6 ~0,1~. Proof. In the words we are able to perform the following reduction procedures: No powers of r, u, m will appear because they are invo- lutory. The generators s and v will occur only as s, s 2, v, 2 v since their order is 3. In every product involving r, s, u, v, we shift r and u to the left by means of sr=rs 2 and vu=uv 2 and in view of r and s commuting with u and v. Negative powers of I t and 2 t will be avoided by means of t~l=(ru)-ltiru=rutiru. Any factors r, u to the immediate right of m will be shifted to the left by means of mr=rm and mu=um. The assertion is now proven. Definition. The number of factors pme6t~ in a collineation is called the stage of the collineation. It is assumed that in each such pro- duct at least one of e and ~ is 1. Moreover, a product (~jm6j)(Pj_ I 6j_Iti) is to be counted as one factor only. A collineation ~ (r,s,u,v~ is defined to be of stage O. P_~roposition 7. Every GT of stage k ~ 0 can be obtained from the fundamental triple by a collineation of stage ~ k. Proof by indt~ion on k. The 56 ordered GT's of Proposition 1 are exactly those produced by the 36 elementB of the group (r,s,u,v~, the direct product of the two groups (r,s~ and (u,v~, both isomorphic to 3. S Steps II and III of stage 1 correspond to the collineations I t and t2: just as each of the new tangents produced just two GT's so do I t and 2. t Step IV corresponds to m6: there are 9 distinct elements G 6 (s,v~ and thus 9 collineations m6. Finally, steps V and VI correspond to the 36 elements ~E ~r,s,u,v~. Thus, if in ~lm~6t~Po e=~=O, we get 36o2"9"36 elements. If e=q, ~=O, we have 36"2.36 elements. For e=O, ~=1, we have only 9-36 collineations because in Pom6Pl the factors u and r in 6~ 1 can be shifted to the left of m so that only elements of the type pm6 remain. The total is then
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