Differential Geometry Peñíscola 1985: Proceedings of the 2nd International Symposium held at Peñíscola, Spain, June 2–9, 1985 PDF
Preview Differential Geometry Peñíscola 1985: Proceedings of the 2nd International Symposium held at Peñíscola, Spain, June 2–9, 1985
Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 9021 laitnereffiD Geometry alocsfSeP 1985 Proceedings of the 2 dn International Symposium held at Pe6fscola, Spain, June 2-9, 1985 Edited by M. A. Naveira, A. Ferr&ndez and .F Mascar6 galreV-regnirpS Heidelberg Berlin NewYork London Paris oykoT Editors Antonio M. Naveira Angel Ferr&ndez Francisca Mascar5 Departamento de Geometrfa y Topologia, Facultad de Matem&ticas Burjasot, Valencia, Spain Mathematics Subject Classification (1980): 22 XX, E 53 XX, A 53 XX, B 53 C XX, 57DXX, 58AXX, 58EXX, 58GXX ISBN 3-540-16801-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16801-X Springer-Verlag New York Berlin Heidelberg sihT work si tcejbus to .thgirypoc llA sthgir era ,devreser rehtehw eht of part or whole eht lairetam si ,denrecnoc yllacificeps of those ,noitalsnart ,gnitnirper esu-er of ,snoitartsulli ,gnitsacdaorb noitcudorper yb gniypocotohp enihcam or ralimis ,snaem dna egarots ni data .sknab rednU 54 § of the namreG Copyright waL are copies where edam than other for etavirp ,esu fee a si elbayap to tfahcsllesegsgnutrewreV" Wort", .hcinuM © gatreV-regnirpS Heidelberg Berlin 6891 detnirP ni ynamreG gnitnirP dna binding: suahkcurD ,ztleB .rtsgreB/hcabsmeH 012345-0413/6412 PREFACE ehT I st International muisopmyS no Differential Geometry was held in Pefiiscola in October, 1982, dna the Proceedings were published in the Lecture Notes in Mathematics, .oN 1045. euD to the interest raised in the mathematical community we get encouraged to continue with periodical meetings no Differential Geometry dna related topics. Following this line we have decided to organize the 2 dn International muisopmyS no Differential also Geometry held in June Pefitscola, 2-9, .5891 This Symposium was attended by approximately seventy mathematicians from all over the world. ehT present volume includes the texts of most of the contributions presented covering several areas of Differential Geometry dna related topics; like nalnnamelR manifolds dna submanifolds, Hermltlan dna Kaehlerlan manifolds, symplectlc dna contact structures, foliations dna analysis no manifolds. ehT editors regret that eud to a general editorial requirement of homogeneity in a Lecture volume, Notes it was not possible to Include other Interesting contributions. All papers have been examined by referees dna we want to thank them for their valuable task. eW would like to thank to all the following Institutions which have contributed to the success of the meeting with generous financial support: Ministerio ed n6lcacudE y Clencta ,).T.Y.C.I.A.C( Conselleria ed Cultura, n6lcacudE y Ciencia ed la dadinumoC amon6tuA Valenciana, Facultad ed Cienclas Matem~ticas ed la Universidad ed Valencia, Dlputaclones Provinclales of Valencia dna Castell6n dna Cultural Services of the hcnerF yssabmE in Madrid. eW thank also to the Instltuto ed Estudios ed Admlnistraci6n Local ed Pefilscola for allowing su to esu their premises during this Symposium, dna Pefilscola's licnuoC. for their kind hospitality with the organizing committee dna with all the participants dna Yturralde, J.M. author of the poster of the meeting. Finally, we would like to acknowdledge the colaboration of all the members of the "Departamento ed y Topologia" Geometria of the Unlversidad ed Valencia. .lirpA 1986 The .srotidE LIST OF PARTICIPANTS .E Abbena .M zednAnreF .U Torino, ylatI .U ,ogBitna8 niapS .V Aldaya .A zedn6rreF .U ,aicnelaV Spain .U ,aicnelaV niapS .5` .I Andersson .E ogellaO .U ,grobeti(8 nedewS, A.U niapS,anolecraB .T Aubin .S 8arbiero .U Paris ,IV France .U ,oniroT Italy .A Asada .O Oil- onardeM .U 6insyu, Japan .U ,aicnelaV niapS .L.R Bishop ,S. tolliO .U ,sionillI U.&A. .U Paris ,lV France R.A BIumenthal .V.V 6oldberg .U .tS Louis, .A.S.U New Jersey .tsnI fo ,.hceT .A.S.U N Bokan .C.d zelazno6 .U Beograd, aivalsoguY .U La Spain Laguna, .E ibalaC .A Gray .U Pennsylvania, .A.S.U .U Maryland, .A.S.U .J.F Carreras d. Orifone .U ,aicnelaV Spain .U Toulouse, France .Y.B Chen .K Orove .U Michigan, .A.S.U .U Maryland, U.&A .D Chinea .L Oualandri .U LB Spain Laguna, .U ,ainoloB ylatI .hP Delano~ .O psauO .U France Nice, .A.U Barcelona, Spain M. Leon de M.I'. Iglesias .U ,agaitna5 Spain .U ,ogaitnaS, Spain .H Donnelly .I Kashiwada Purdue ,.U .A.S.U .U Ochanomizu, Japan .J.F Echarte .5.R Kulkarni .U ,alliveS Spain .U ,anaidnI .A.S.U M. illeticlaF .A Lichnerowicz .U Bari, ylatI e~ClloC France France, de lV .M zednanreF Andres .0 uloiccapuL .U Seville, niapS .U Rome, ylatI .M.L Fernandez FernYndez Llabres M. .U ,allive6 6pain .A.U 6pain Barcelona, A Machado .F.J Pommaret .U Lisboa, Portugal .cE .taN Pants te ,.hC France Paris, .Y Maada .H.A Rocemora .U Japan Keio, .U ,aicnelaV $pam Margalef J.M. zeugirI(oR .B .C.I.S.C Spain Madrid, .U ,ogaitnaS Spare .F Marhuenda .A soR .U Spain Valencia, .U ,adanarO erapS .Y Marina .C Ruiz .U ,airbalaC ylatI .U ,adanarG erapS .A.R Marinosci M.R. odagI%, .U Lecce, Italy .U Santiago, erapS .F Mascer6 .S ~gura .U ,aicnelaV Spain .U ,aicnelaV erapS .S.R Millman .K awagikeS ,.F.S.N Washington, .A.6.U .U ,atagiiN napaJ .Y Miquel Sakizawe M. .U ,aicnelaV Spain .U Tokyo Oakugei, Japan d. Monterde .R Sulanke .U Spain Valencia, Humboldt ,.U East Berlin, Oermany .A Montesinos .hP londeur .U ,aicnelaV Spain .U ,sionillI .A.S.U .5 Montiel .F.J Lopera Torras .U Oranada, Spain .U ,ogaitnaS Spain Naveira A.M. .F Ur bano .U ,aicnelaV Spain .U ,adanarO niapS ,A Pastor .L ekcehnaV .U Valencia, Spain .U ,nevueL muigleB Pastore A.M. .F aleraV .U Bari, ylatI .U Murcia, niapS P.Piccinni E.H repmeknlekniW .U Rome, Italy .U Maryland, .A.6.U ELBAT FO STNETNOC .B.5 ,REDNAXELA .D.I OREB dna .L.R .POHSIB uniqueness Cauchy in the Riemannian obstacle ............................................problem ............................................................................... 1 .1.5 NOSSREDNA Non-abelian egdoH theory via heat flow .......................................................... 8 .A .ADASA noN Abelian Poincar6 Lemma ................................................................................... 37 .T .NIBUA eL ProblGme ed ebamaY concernant la courbure scalaire .................................... 66 .M SORRAB dna .Y.B .NEHC Finite type spherical submanifolds ............................................. 73 .D.I GREB eeS . B.S .REDNAXELA RL. .POHSIB eeS . B.S .REDNAXELA .A.R LAHTNEMULB between Mappings manifolds with Cartan connections ........................ 94 J.P. .NONOIUGRUOB Invariants intGgraux fonctionnels pour sed equations d derivGes aux partielles d'origine euqirtGmoGg .................................................................................... I00 .J.F ,5ARERRAC ZEDNARREF.A dna .V .LEUQIM Hermitian natural differential operators.. 901 .Y.B NEHC eeS .1"I 5ORRAB .D AENIHC dna .C .ZELAZNOO An example of na almost cosymplectic suoenegomoh manifold ................................................................................................................................ 331 .H .YLLENNOD Positive solutions of the heat dna eigenvalue equations no Riemannian manifolds ........................................................................................................................... 341 .J.F .ETRAHCE Etude sed algGbres ed Lie rGsolubles rGelles qui admettent sed xua'edi unidimensionels n'appartenant sap ua centre ............................................................ 251 .M ZEDNANREF .dAna .YARG The Iwasawa manifold ................................................................. 751 .A .ZEDNARREF eeS .J.F SARERRAC .O LIO .ONARDEM sums Connected dna the infimum of the ebamaY functional 061 .................. .V.V .OREBDLOG Isoclinic webs W(4,2,2) of maximum 2-rank ............................................... 861 .C .ZELAZNOO eeS .D AENIHC VIII .A .YARG eeS .M ZEDNANREF .W.F ,REBMAK .A,E HUR dna .hP .RUEDNOT Almost transversa]]y symmetric foliations... 184 .S.R dna INRAKLUK .U .LLAKNIP Uniformization of geometric structures with aplications to conformal geometry 190 ........................................................................................................... .A LICHNEROWICZ. quotient Repr6sentation coadjointe te espaces homogenes de ..........................................................................................................contact ..................... 210 .V MARINO and .A PRASTARO. On a geometric generalization fo the Noether ...meroeht 222 .V MIQUEL See .J.F CARRERAS .M.A NAVEIRA dna .S .ARUGES ehT isoperimetric inequality dna the geodesic spheres. emoS geometric secneuqesnoc ....................................................................................... 235 .U .LLAKNIP eeS .S.R .INRAKLUK .A .ORATSARP eeS .V .ONIRAM .F .RETHCIR nO the k-dimensional nodaR transform no rapidly decreasing functions... 243 .A .SOR Kaehler submanifolds in the complex projective space ............................................... 259 .A.E .HUR eeS .W.F .REBMAK .S .ARUGES eeS .M.A .ARIEVAN K AWAGIKES dna .EKCEH N.ALV Volume-preserving geodesic symmetries no four dimensional Kaehler manifolds ..................................................................................... 275 .hP .RUEDNOT eeS .W.F .REBMAK .F.J SERROT .AREPOL ehT cohomology dna geometry of Heisenberg-Reiter nilmanifolds ..................................................................................................................... 292 .F .ONABRU Totally real submanifolds of a complex projective space ............................ 302 .L .EKCEHNAV eeS .K .AWAGIKES CAU~ U N I ~ IN THE RIE~IAN OBSTACLE PRORI~ Stephanie B. Alexander I. David Berg Richard L. Bishop Department of Mathematics University of Illinois 1409 West Green Street Urbana, Illinois 61801 §I. Introduction. In a Rie~nnian manifold-with-boundary it is not generally true that geodesics (locally shortest paths) have the Cauchy uniqueness property. For example, whenever there is a boundary direction in which the boundary bends away from the interior, there will be an obvious one-parameter family o£ distinct geodesics o£ a given sufficiently small length which start in that direction. Two geodesics of such a family coincide on an initial segment, after which one 0£ them is a geodesic o1 the interior. The following theorem states that this is the only manner in which Cauchy uniqueness fails. By an involute of a geodesic D is meant a geodesic which has the same initial point, initial tangent vector, and length as ~, and which consists of a maximal initial segment in common with ~ followed by a nontrivial geodesic segment of the interior. Theorem 1 (Cauehy uniqueness for manifolds-with-boundary). Every boundary point has a neighborhood in which: if two geodesic segments with the same initial point, initial tangent vector and length do not coincide, then one of them has its right endpoint in the interior and is an involute o£ the other. In studying the geodesics of Rieraannian manifolds-with-boundary, we are studying, for example, the geometry of wave£ront propagation around an obstacle in an isotropic medium (since the orthogonal trajectories of the wavefronts are geodesics in the appropriate Riemannian manifold-with-boundary). In another approach to the analysis of bifurcation o£ geodesics. Arnol'd has studied the singularities of wave fronts generated by obstacles in general position [A]; for an extensive bibliography, see [ABB]. Here we are interested in what controls the tendency of geodesics emanating from a boundary point, sometimes bearing on the boundary and sometimes travelling through the interior, to pull apart from one another. eW deal throughout with a ~ C Riemannian manifold-with-boundary ,M with ~ C boundary B. In this setting, the geodesics cannot be described by differential equations with Lipschitz continuous coefficients. Our problem, that of analyzing how the geodesics of H are controlled by the interaction between the geodesic equations of the boundary and the interior, is one to which routine techniques do not apply. In [ABB]. it was shown that M cannot have "positively infinite curvature" at the boundary. Specifically, an estimate was found for the least distance at which two geodesics emanating from a point p can rejoin (i.e., the cut radius of M), and the rate at which geodesics from p can pull together. The estimate is in terms of the tubular radius of .M namely the supremum of all R for which M can be imbedded in some Euclidean space so that every point at distance R or less from M is the center of a closed ball which meets M at a single point. This single invariant reflects upper bounds of curvature and lower bounds of cut radius for both the interior and the boundary of .M On the other hand. M can certainly have "negatively infinite curvature" at the boundary, in the above sense of having a family of geodesics with the same initial tangent. However, our proof o£ Theorem 1 has the following consequence. For any C > 1 there is a p > 0 depending only on C and the tubular radius o£ M such that: if the endpoint of a geodesic from any point p is moved along the boundary at a fixed distance s < p from p, then the ratio of the endpoint separation to the initial tangent separation is bounded above by Cs. In this sense, we have established integral bounds for the tangential curvature at a boundary point. No assumption is made on the boundary except smoothness. If we were to assume that every geodesic segment had finitely many switchpoints (points at which it switches from nontrivial boundary segment to nontrivial interior segment}, then Cauchy uniqueness would be straightforward. However, besides boundary segments, interior segments and switchpoints, a geodesic may contain accumulation points of switch points, which we call intermittent points. Indeed, it is easy to construct a geodesic segment containing a set of positive measure of intermittent points. When intermittent points are allowed, the proof of Cauchy uniqueness becomes quite delicate. Intermittent points are not rare, in the sense that, for example, arbitrarily close in the 2 C norm to any negatively curved surface in 3 E is a surface for which the manifold-with-boundary lying to one side has geodesics with intermittent points. On the other hand, manifolds-with-boundary whose geodesics have intermittent points are apparently not generic in any reasonable sense. An important motivation for studying the general case is that this seems to be the most direct way to obtain curvature bounds, independent of the number and behavior of switchpoints, even in the generic case. It is shown in [ABB] that every point of B has a neighborhood in which no involute of any geodesic segment can end on B. (This is because locally an involute of a geodesic 9 lies above ~ with respect to the inward normals to B.) Therefore Theorem 1 is an i~ediate consequence of the following statement: {m) two geodesics with the same initial tangent vector at p must coincide on an interval if each geodesic touches B arbitrarily close to p. The proof of claim (m} will use the following regularity theorem. By a collar neighborhood for B is meant a neighborhood of some boundary point which is foliated by interior geodesics normal to B. Theorem 2. [ABB] Any geodesic ~ of M is C .I The acceleration of exists except at the (countably many) switchpolnts, and vanishes at the intermittent points. If D lies in a collar neighborhood for B, then its normal projection to B is 2 C with locally Lipschitz second derivative. .2§ Proof of claim (~). On a collar neighborhood for B at p, consider coordinates n x ..... I x adapted to B: that is, let n x be the distance from B, and let the k x for k < n be arbitrary coordinates on B which are extended to be constant on the geodesics normal to B. Then the equation of a geodesic D of M has the following form ([ABB]): ~x =- ~i.j ~x ]x rlj k (1) (2) n'X =- z~- n<J,l~ Fij .n x~ x] Here the Fij k are the connection coefficients of the interior of M, and K~ is defined on the boundary segments of ~ to be the normal curvature of B in the direction of ~', and at all other points of ~ to be O. (1) holds everywhere, and (2) holds except at switchpoints. (For instance, on boundary segments the right- hand side of (2) vanishes, as required, because -Fij n is the second fundamental form of B.) Since the Fnn k vanish, we may rewrite (I) as follows: (3) Xtr n = Fk(Xl .... . Xn'Xl'" # •. 'Xn-l) t + 'x n %(Xl' ... 'Xn'Xl 'Xn-l.... ) ' ' where the k F and k G are ~ C in their 2n - 1 arguments. Suppose ~ and T are geodesics with the same initial tangent vector at p. Set ~i = xi o ~, Ti = xi o T, X = (T1 - ~1 ..... Tn-1 - ~n-1 )' Y = X', z = Tn -~n' F = (F 1 ..... Fn_l), and G = (G 1 ..... Gn_l). Then (3) gives
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