Table Of ContentLecture 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
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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
