Selected Papers -Chuan-Chih Hsiung
Chuan-Chih Hsiung
World Scientific
Selected Papers of Chuan-Chih Hsiung
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Selected Papers -Chuan-Chih Hsiung
Chuan-Chih Hsiung
World Scientific
Selected Papers of Chuan-Chih Hsiung
Selected Papers of Chuan-Chih Hsiung
Chuan-Chih Hsiung Lehigh University, USA
V f e World Scientific wb
Singapore *• New Jersey • L London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
SELECTED PAPERS OF CHUAN-CHIH HSIUNG Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-02-4323-5
Printed in Singapore by Uto-Print
**™*^
-jr
\ CHUAN-CHIH HSIUNG
^
To my wife Wenchin Yu and my daughter Nancy for their continuous love and support
IX
PREFACE
During the period 1935-1943 I was at the National Chekiang University in Hangchow, China working on projective differential geometry under Professor Buchin Su. I then came to the United States in 1946, where I started to change gradually to work on global problems. Altogether, I have published about one hundred research papers, from which I have selected sixty four chronlogically in this volume. I am very happy to have this book recording my research, and I further hope it can also be used as a reference by other people. I wish to express my thanks to the World Scientific Publishing for their cooperation and help in publishing this volume. CHUAN-CHIH HSIUNG Bethlehem, Pennsylvania, U.S.A. November, 1999
XI
P E R S O N A L A N D PROFESSIONAL HISTORY
For many generations, my ancestors lived in a small village called Shefong, which is about ten miles from Nanchang, the capital of province of Jiangxi. The village was located in a picturesque setting, bordered by a shimmering lake in the front and short blue-shaded mountains in the rear. My ancestors were all farmers until my grandfather's generation when his only brother and he studied the works of Confucius and other literary scholars. Unfortunately, they died within two years of each other, both in their mid-thirties. At that time, my father (Mu-Han), the second of three sons, was only eight years old. Fortuitously, a distant uncle considered my father to be especially intelligent and decided to give him the education that my grandfather had received. As was predicted, my father excelled in his studies and passed the civil examination of his district before the age of twenty. Normally, the next step would have been to take the civil examination of the province; however, that year, the entire examination system was changed and the western school system was adopted. Each province established an "Advanced School"; my father attended the Advanced School in Nanchang. The new school was composed of two classes: 1) First class for the study of literature and law and 2) Second class for the study of mathematics and physics. Although having already passed the civil examination, which showed that he was proficient in literature, he should have entered the first class. However, he decided to enroll in the second class to study mathematics, which was taught using English textbooks. He enjoyed the challenge and, four years later, not only graduated with high honors but also was appointed to be the vice principal and to teach mathematics at the new Jiangxi Provincial First High School. After I grew up, I had a chance to look at his early schoolwork. I was impressed by his precise handwriting, as demonstrated in his plane geometry reports; his figures that were drawn using a ruler and compass were of comparable quality to those that were printed. My father was an early influence on my profound interest in mathematics. My father and mother (Tu Shih) had four sons and one daughter. I was the third of four sons; my sister was the youngest child. My father was a very good teacher and taught high school mathematics to my two older brothers and me. He gave me a good foundation in mathematics and initiated the attraction that has remained a lifelong companion. My second oldest brother, C.Y., is also a professor of mathematics; he taught at Wuhan University and has written several college textbooks in algebra and theory of numbers.
xii
I was born on February 15, 1916 in the village of Shefong. When I was three years old, my family with my grandmother moved from the village to Nanchang, where I grew up. In 1932 I graduated from the First Jiangxi Provincial High School in Nanchang. At that time in China there were only a few good universities. So in order to enter one of those universities one had to pass a very competitive national-wide entrance examination, I was fortunate to be admitted to the National Chekiang University in Hangchow majoring in mathematics, so I could study under Professor Buchin Su. After my graduation in 1936, because of the war in China I was not able to come immediately to the U.S.A. for further study. Although Professor V. Guy Grove at Michigan State University helped me to get a teaching assistantship in 1943, I could not arrive on campus until 1946. In 1948, I received my Ph.D. from Michigan State and have the distinction of being the Michigan State University's first Ph.D. in mathematics. Following my graduation, I was an instructor at the University of Wisconsin in Madison for two years, followed by a quarter as a visiting lecturer at Northwestern University. On Professor Hassler Whitney's recommendation, I took the opportunity to be a research fellow with him at Harvard University, until he moved to the Institute for Advanced Study at Princeton in the fall of 1952. I benefited very much from my visit to Harvard; I was able to learn the latest developments in mathematics. Since 1952, I have been a professor at Lehigh University. In the course of my research, I realized that there was a genuine need for a special journal dedicated to the field of differential geometry. Therefore, in 1967, with the support of Lehigh University I founded and published the Journal of Differential Geometry. Under the influence of this journal, differential geometry has become a very active branch of mathematics, with a scope far exceeding its former classical one.
Xlll
TABLE OF CONTENTS
Preface
ix
Personal and Professional History
xi
* [4] Sopra il Contatto di Due Curve Piane
1
[8] (with F. T. Wang) A Theorem on the Tangram
10
[10] Projective Differential Geometry of a Pair of Plane Curves . .
14
[12] An Invariant of Intersection of Two Surfaces
22
[13] Projective Invariants of a Pair of Surfaces
26
[15] Projective Invariants of Intersection of Certain Pairs of Surfaces
30
[16] Some Invariants of Certain Pairs of Hypersurfaces [18] A Projective Invariant of a Certain Pair of Surfaces
35 . . . .
[19] Projective Invariants of Contact of Two Curves in Space of n Dimensions [23] On Triplets of Plane Curvilinear Elements with a Common Singular Point
46
49 56
[27] Invariants of Intersection of Certain Pairs of Curves in n-Dimensional Space
60
[28] Affine Invariants of a Pair of Hypersurfaces
69
[33] Some Integral Formulas for Closed Hypersurfaces
73
[34] A Theorem on Surfaces with a Closed Boundary
82
[35] On Differential Geometry of Hypersurfaces in the Large . . .
88
[37] Some Global Theorems on Hypersurfaces
98
Numbers in brackets refer to the Bibliography on pp. 685-692.
[38] A Uniqueness Theorem for Minkowski's Problem for Convex Surfaces with Boundary
108
[39] Curvature and Betti Numbers of Compact Riemannian Manifolds with Boundary
113
[40] A Uniqueness Theorem on Two-Dimensional Riemannian Manifolds with Boundary
150
[42] (with S. S. Chern and J. Hano) A Uniqueness Theorem on Closed Convex Hypersurfaces in Euclidean Space
156
[43] Some Uniqueness Theorem on Riemannian Manifolds with Boundary
160
[44] Isoperimetic Inequalities for Two-Dimensional Riemannian Manifolds with Boundary
175
[45] A Note of Correction
183
[46] (with S. S. Chern) On the Isometry of Compact Submanifolds in Euclidean Space
186
[47] Curvature and Homology of Riemannian Manifolds with Boundary
194
[48] Vector Fields and Infinitesimal Transformations on Riemannian Manifolds with Boundary
209
[49] On the Congruence of Hypersurfaces
233
[51] On the Group of Conformal Transformations of a Compact Riemannian Manifold [52] Structures and Operators on Almost-Hermitian Manifolds [54] (with J. K. Shahin) Affine Differential Geometry of Closed Hypersurfaces [55] On the Group of Conformal Transformations of a Compact Riemannian Manifold. II [56] (with Y. K. Cheung) Curvature and Characteristic Classes of Compact Riemannian Manifolds
242 . .
247
264 285 290
XV
[57] (with J. D. Liu) The Group of Conformal Transformations of a Compact Riemannian Manifold
299
[58] (with B. H. Rhodes) Isometries of Compact Submanifolds of a Riemannian Manifold
305
[59] On the Group of Conformal Transformations of a Compact Riemannian Manifold. Ill
321
[60] (with S. Braidi) Submanifolds of Spheres
327
[61] Minimal Immersions in Riemannian Spheres
344
[63] (with J. J. Levko III) Curvature and Characteristic Classes of Compact Pseudo-Riemannian Manifolds
351
[64] (with J. J. Levko III) Complex Laplacians on AlmostHermitian Manifolds
365
[65] (with L. W. Stern) Conformality and Isometry of Riemannian Manifolds to Spheres
386
[66] (with S. S. Mittra) Isometries of Compact Hypersurfaces with Boundary in a Riemannian Space
395
[67] (with L. R. Mugridge) Riemannian Manifolds Admitting Certain Conformal Changes of Metric
412
[68] The Signature and G-Signature of Manifolds with Boundary .
421
[69] (with L. L. Ackler) Isometry of Riemannian Manifolds to Spheres [70] (with F. Brickell) The Total Absolute Curvature of Closed
425
Curves in Riemannian Manifolds [71] A Remark on Pinched Manifolds with Boundary [72] The Generalized Poincare Conjecture on Higher Dimensional Manifolds with Boundary [73] (with T. P. Lo) Congruence Theorems for Compact Hypersurfaces of a Riemannian Manifold
437 454
459 464
XVI
[74] (with N. H. Ackerman) Isometry of Riemannian Manifolds to Spheres. II [75] A Remark on Cobordism of Manifolds with Boundary
480 . . .
[77] (with J. D. Liu and S. S. Mittra) Integral Formulas for Closed Submanifolds of a Riemannian Manifold [78] (with J. D. Liu) A Generalization of the Rigidity Theorem of Cohn-Vossen
490
495 514
[79] (with L. R. Mugridge) Euclidean and Conformal Invariants of Submanifolds
523
[82] (with K. S. Park) Some Uniqueness Theorems on Two-Dimensional Riemannian Manifolds Immersed in a General Euclidean Space
531
[83] Nonexistence of a Complex Structure on the Six-Sphere . . .
548
[84] (with K. M. Shiskowski) Euler-Poincare Characteristic and Higher Order Sectional Curvature. I [85] (with J. J. Levko) Conformal Invariants of Submanifolds. II
565 .
581
[86] (with L. Friedland) A Certain Class of Almost Hermitian Manifolds
590
[87] (with B. Xiong) A New Class of Almost Complex Structures
602
[88] (with C. X. Wu) The Spectral Geometry of Almost L Manifolds [89] (with W. Yang and L. Friedland) Holomorphic Sectional and Bisectional Curvatures of Almost Hermitian Manifolds
619 .
632
[90] (with W. Yang and B. Xiong) The Spectral Geometry of Some Almost Hermitian Manifolds
654
[91] Some Conditions for a Complex Structure [92] Nonexistence of a Complex Structure on the Six-Sphere. II Remarks on Some Selected Papers
670 .
673 679
XV11
Curriculum Vitae
683
Bibliography of the Publications of C. C. Hsiung
685
List of Ph.D. Theses Written Under the Supervision of C. C. Hsiung
693
Permission
695
Sopra il contatto di due curve plane. Nota di
CHTJAX-CHICH HSIUITO
(Ishan, Kwangsi, China).
Sunto. - L'Autore da la costruzione geometrica delle fette r' ' considerate in una ricerca di BOMPIANI per k = 2, 3 e 4. 1. E ben noto che se due curve piane C e C hanno un contatto d' ordine u. — 1 (p > 1) nel loro punto comune 0 in modo che gli sviluppi che rappresentano queste curve nelia vicinanza di 0 siano lispettivamente y = ax^ -+-...,
y = axil-h...
(a^=a), a 11 rapporto -=- e un invariante proiettivo. Di esso C. SEGBE (J) ha a dato la seguente interpretazione geometrica: Si consideri una xetta r prossima ad 0 secante i rami considerati e la tangente comune in 0 risp. nei punti P, P, T; e sia M un punto qualsiasi di r. Allora, quando r teude ad 0, il limite del birapporto (PP, TM) a e -=-, a condizione che la posizione limite della trasversale r e a del punto M siano rispettivamente diversi dalla tangente comune e da 0. II prof. E. BOMPIANI (') ha ripreso questa ricerca e ha determinato la posizione limite r\ della trasversale r per curve C e C che hanno un contatto d' ordine k (intero) in 0 affinche s' annulli il termine d' ordine k nello sviluppo del birapporto (PP, TM); ed ha anche ottenuto una costruzione geometrica per la retta r\ (l) C. SEGRE, SU alcuni punti singolari delle curve algebriche e sulla tinea parabolica di una superficie, « Bend. B. Ace. Lincei », vol. "VI. serie V (1897), pp. 168-175; Sugli elementi curvilinei che hanno comuni la tangente •e il piano osculatore, « Bend. B. Ace. Lincei », vol. XXXIII, serie V (1924), l>p. 325-329. (*) E. BOMPIANI, Invarianti proiettivi di contatto fra curve piane, « Bend. B. Ace. Lincei >, vol. I l l , serie VI (1925), pp. 118-123.
444
CHUAN-CHICH HSIUNG
Sembra piuttosto difficile dare una costruzione geometrica p e r la retta r 0
quando k > 1. L' autore e riuscito soltanto a costruire
le rette r\ , r 0 Nota.
e r 0 , ci6 che forma appunto il contenuto di questa.
2. Supponiamo che le due curve C e C abbiano un contatto d' ordine k (k > 1) in u n punto ordinario 0 (semplice e non di flesso) colla tangente comune t. Se prendiamo 0 come origine (0, 0) e f come asse delle x in u n sistema di coordinate cartesiane, le d u e curve nella vicinanza di 0 sono rappresentate da y = xi 2 a(x*',
C: (1) / C: [
y=
xtZaixi, i=0
in cui a, = a,, (* -f-1 < k), a*_x ={= a*_, Supponiamo che l'equazione della trasversale r sia (2) x = a(E)t/ -+- e, dove a0 = a(0) sia finito. Quando | E | e piccolissimo, 1' ascissa del punto d'intersezione P di r e C pu6 essere espressa da una eerie di potenze in E: (3)
X = e(l -l- ae 2 S^'), ;=« ossia, per l'ordinata di JR, t/ = s' 2 8^>.
(4)
Siccome P deve trovarsi sulla curva C, abbiamo che (5)
2 S,E->' = (1 -f- ae S S^)' 2 a;E'(l -(- as 2 8^)* ;'=0
;=0
«=0
;=0
Eguagliando i coefficienti dei termini simili nei due membri della (5), possiamo determinare le 8 ( . Si vede cosl che qualsiasi o, pub venire espresso soltanto con a, (i<j). Troviamo in particolare che IK
= r «o>
k [ U
= «! •+• 2aoO.S = «! •+- 5a 0 o 0 a, -4- BKO'V + 2a0, S a o> 1 = a, •+• 6x 0 a 0 a, -f- 3x0a, -t-21ac, V « i -+ •+• 1 4 * V ' + 10a0*0 0 V - 2a0V, a = t •+• 7a0o,a0 -+- ...
s
(6)
Ik'
5a 0 o 0 Oi
3
445
SOPRA IL CONTATTO DI DUE CURVE PIANE
N a t u r a l m e n t e le r e l a z i o n i c o r r i s p o n d e n t i p e r l a c u r v a C si ot~ t e n g o n o dalle (6) sostituendo r i s p e t t i v a m e n t e a,, 8,- con a,-, 8( (j •+• 1 i > ft)- P e r c o n s e g u e n z a 1' ascissa del p u n t o d' i n t e r s e z i o n e P" di r e C e
PI
ac = s(l -+- as 2 8-s'), >=o
i n cui
s, = s,
y +. i < *).
Consideriamo ora sulla t r a s v e r s a l e r u n p u n t o q u a l s i a s i M(x0, y0),. p u r e h e p e r s = 0 sia M^=0; il b i r a p p o r t o dei q u a t t r o p u n t i P , P , T, M e 2 8 ^ x 0 — e(l -f- as S 8^) (8)
D = {PP, TM) = (xx, ix0) =
;=0
J'=0"
S 3,-sJ x„ — s(l + « s S S,s->') j=0
1 +-
(8t-1~8t-,)+(St-St)«+(2) 1 °°1 -f- — S S,e'-
;=0 rl(8,_1-8,_1)+(S,-8>-.(2)" oo
Xj — s(l -+- ae E Sj-e-')
«0;=0
i n cui (n) i n d i c a t u t t i i t e r m i n i di grado > D a l l a (8) si r i c a v a che (9)
log D =
(5
*-' ~
S
n i n e.
~'- l } . * - i +
•+• ^ K f e - ^ - ^ - . - V i ) ! « » - 1 ( S ' " ~ 2 8 i ~ l } t «"*-» + (ft +1). Si p u 6 d e t e r m i n a r e a0 a n n u l l a n d o il coefficiente di s* n e l l a (9). Per- A; = 2 si o t t i e n e (10)
6a 0 2 (a,— a x )
K * - « i * ) — 2 O 0 ( » J —a,)],
p e r A; = 3 1
— [a,(a, - a,) - aa(a3 - a 3 )], 4o0*(ax — o . )
(11) e p e r ft = 4
1
a„ =
5a 0 -(a, — a 3 )
[«i(a» — a 3 ) - a 0 (a 4 — at)].
P e r c o n s e g u e n z a le equazioni delle r e t t e p r o i e t t i v a m e n t e v a r i a n t i r0, r 0 ( , ) e r 0 (4) sono r i s p e t t i v a m e n t e : (12)
1 6a 0 *(a, — a,)
[2a 0 (a, - a,) -
(a,' -
afljy
= 0,
co-
446
(13) (13)
CHTJAN-CHICH HSIUNTG
X H
[a0(at - a3) — a ^ — at)]y == 0, 4a 0 s (a, - a2) 1 [ a , , ^ — a4) — o,(o, - at))y = 0. 5a08(as - ««)
3. Anzitutto eonsideriamo il caso k = 2. Siano rispettivamente C, e Cs le coniche osculatrici in 0 di C e di C. Per la (1) 1' equazione di C, e (14)
ajx1 -t- a ^ -+• ^
- ^ j * / 5 — a 0 # = 0,
e similmente 1'equazione di C, (15)
o 0 V -t- a,xy -+- fe. - ^ f V - a0y = 0. \«o «o I "Vi sono oo< cubiche C3, ciascuna delle quali ha contatto 5-punto con la curva C in 0. Per la (1) 1' equazione di una C3 e
(16)
(a0G •+• a,)a;8 H
(a^G- -+- a0'H associata in O alle curve C e C. 5. Per trovare una costruzione geometrica della retta covariante r0(4)? consideriamo le cubiche C,, C, che hanno in 0 con-
SOPRA H J CONTATTO DI DUE CUEVE PIANE
449
"tatto 7-punto risp. con le curve C, C; le loro equazioni sono rispettivamente (31)
(a0G -+- a,)*' •+- — (aft -+- a0'H -+- at)x*y -+•+• TTTlKaj — a,J)G -+- ao'a.il-t- {a0a.A - a,a,)]a;y8 -+5 [(a0!aJ ~ 3a 0 aiaj + 2a18)G •+• a 0 x ( a o a J — ai*W +-
H a
o
-i- (a^cit — 2a 0 a 1 a, — a0a2* -+- 2aiia^j\y3 -+- ao** — Gxy — -HJ/' — y =. 0, e (32) (aaG -+- ajx3 H (a,G -+- a^H-*- ajx-y •+• •+• TTs [ K « t — «/)& •+• a*aJi-*-
K«3 — OiO*)]^* •+-
-+- in. [(ao% — 3a 0 a,a, -f- 2a l ')G -+- a08(aoaj — a,2)ir-+-+- {a^ai — ^oaiai
— a o a j ! •+• 2»1!a2)]?/3 -4- a 0 x : — Gxy — Hy* — y = 0.
dove G, H; G, H sono costanti arbitrarie. Un facile calcolo mostra che il sistema delle rette congiungenti 0 alle intersezioni =|= 0 di C, e C3 e (33)
aa(ai-a3)(G-G)x*
+ [(G-G)\(a