Selected Papers of Chuan-Chih Hsiung

Preface......Page 10
Personal and Professional History......Page 12
TABLE OF CONTENTS......Page 14
Sopra il Contatto di Due Curve Piane......Page 19
2. Lemmas......Page 28
3. Proof of the theorem......Page 29
4. Remark......Page 31
2. Two projective invariants......Page 32
3. The canonical triangle and geometrical characterizations of the invariants......Page 33
4. Canonical power series expansions......Page 36
5. A generalization of the covariant line of Bompiani......Page 38
2. Derivation......Page 40
3. A projectively geometric characterization......Page 41
4. A metrically geometric characterization......Page 42
2. A projective invariant......Page 44
3. A geometrical characterization of the invariant I......Page 45
4. Certain projective transformations and invariants......Page 46
2. Derivation of invariants.......Page 48
3. Projective characterizations of the invariants I J......Page 49
5. A projective characterization of the invariant I......Page 51
REFERENCES......Page 52
Introduction......Page 53
1. Derivation of an invariant.......Page 54
2. A projective characterization of the invariant I......Page 55
3. A metric characterization of the invariant I......Page 56
4. Derivation of invariants.......Page 59
5. Projective characterizations of the invariants I J......Page 60
6. Metric characterizations of the invariants I J......Page 61
A Projective Invariant of a Certain Pair of Surfaces......Page 64
REFERENCES......Page 66
1. Introduction......Page 67
2. Derivation of invariants......Page 68
3. Geometrical characterizations of the invariants Ji n~i+2......Page 69
4. A geometrical characterization of a general invariant Iij......Page 71
2. Derivation of an invariant......Page 74
3. Geometrical characterizations of the invariant I......Page 75
1. Derivation of Invariants.......Page 78
2. Metric and projective characterizations of a general invariant Ii.......Page 79
3. Derivation of invariants.......Page 81
4. Metric and projective characterizations of general invariants Ii and Ji.......Page 83
2. Derivation of invariants......Page 87
3. Metric and affine characterizations of the invariants......Page 88
REFERENCES......Page 90
Introduction......Page 91
1. Preliminaries......Page 92
2. Proof of the formula (0.2) for r = 0......Page 94
3. Proof of the formula (0.2) for a general r......Page 95
4. Proofs of Theorems 2 and 3......Page 98
References......Page 99
S 2. Preliminaries......Page 100
S 3. An integral formula......Page 102
S 4. Proof of the theorem......Page 103
References......Page 105
2. Preliminaries......Page 106
3. Some integral formulas......Page 110
4. Proof of the theorem......Page 113
REFERENCES......Page 115
1. Introduction......Page 116
2. Preliminaries......Page 117
3. An integral formula......Page 119
4. Proof of Theorems 1 and 3......Page 121
5. Connection with symmetrizations......Page 123
REFERENCES......Page 125
2. Preliminaries......Page 126
3. Proof of the theorem......Page 128
1. Introduction......Page 131
2. Riemannian submanifolds in Euclidean space......Page 133
3. Differential forms and tensors......Page 135
4. Integral formulas......Page 142
5. Harmonic and Killing vector fields......Page 148
6. Harmonic and Killing tensor fields......Page 153
7. Flatness and deviation from flatness......Page 157
2. PRELIMINARIES......Page 168
3. AN INTEGRAL FORMULA......Page 170
4. PROOF OF THE THEOREM......Page 171
REFERENCES......Page 173
A Uniqueness Theorem on Closed Convex Hypersurfaces in Euclidean Space......Page 174
1. Introduction......Page 178
2. Immersed submanifolds in Euclidean space......Page 180
3. Integral formulas for a pair of immersed manifolds with boundary......Page 183
4. Proofs of Theorems I and II......Page 187
5. Integral formulas for convex hypercaps......Page 188
6. Proof of Theorem III......Page 190
REFERENCES......Page 191
2. Preliminaries......Page 193
3. Lemmas......Page 195
4. Theorems......Page 196
A Note of Correction......Page 201
1. Preliminaries on Algebra......Page 204
2. Preliminaries on Differential Geometry and Statement of Theorem......Page 205
3. Integral Formulas......Page 206
4. Proof of the Theorem......Page 209
2. Notations and operators......Page 212
3. Harmonic and Killing tensor fields and local boundary geodesic coordinates......Page 214
4. Integral formulas and theorems......Page 216
5. Existence of a constant symmetric tensor......Page 219
6. Riemannian manifolds with zero Ricci curvature......Page 221
7. A general application of Theorems 4.1 4.2 4.3......Page 222
8. Riemannian manifolds in spaces of constant curvature......Page 223
9. Symmetric manifolds......Page 225
References......Page 226
Introduction......Page 227
1. Notations and operators......Page 228
2. Lie derivatives and infinitesimal transformations......Page 231
3. Local boundary geodesic coordinates and integral formulas......Page 233
4. Killing vector fields and infinitesimal motions......Page 237
5. Conformal Killing vector fields and infinitesimal conformal motions......Page 238
6. Projective Killing vector fields and infinitesimal projective motions......Page 246
REFERENCES......Page 249
1. LEMMAS......Page 251
2. HYPERSURFACES IN EUCLIDEAN SPACE......Page 252
3. INTEGRAL FORMULAS......Page 254
4. THEOREMS......Page 256
2. Notations and Formulas......Page 260
3. A Lemma......Page 262
4. Proof of the Main Theorem......Page 263
Introduction......Page 265
1. Notations and real operators......Page 266
2. Complex structures and operators......Page 268
3. Proof of Theorem 3.1......Page 272
4. Realization of the complex operator []......Page 274
REFERENCES......Page 280
Introduction......Page 282
1. The affine group and its structural equations......Page 283
2. Frenet affine frames......Page 284
4. The canonical expansion and the fundamental forms......Page 288
5. Affine invariants......Page 290
6. Relations between affine and metric invariants......Page 292
7. Integral formulae......Page 294
8. Theorems......Page 296
REFERENCES......Page 302
1. Introduction......Page 303
2. Lemmas......Page 304
3. Proof of Theorem 5......Page 305
REFERENCES......Page 306
1. Higher order sectional curvatures......Page 308
2. Characteristic classes......Page 310
3. Relationships between curvatures and characteristic classes......Page 312
References......Page 316
1. Introduction......Page 317
3. Proof of Theorem 2......Page 318
1. Lemmas......Page 323
2. Submanifolds of a Riemannian manifold......Page 325
3. Integral formulas......Page 329
4. Theorems......Page 333
1. Introduction......Page 339
2. Lemmas......Page 340
3. Proof of Theorem 3......Page 342
References......Page 344
2. Laplacian of the Second Fundamental Form......Page 345
3. Integral Formulas......Page 348
4. Main Theorems......Page 353
1. Introduction......Page 362
2. Notation and formulas......Page 363
3. Proof of Theorem 1.1......Page 365
REFERENCES......Page 368
Introduction......Page 369
1. Fundamental formulas......Page 370
2. Characteristic classes......Page 371
3. Relationships between curvatures and characteristic classes......Page 376
Introduction......Page 383
1. Notation and real operators......Page 384
2. Complex structures and operators......Page 386
3. Expressions for []'s......Page 393
4. Realization of []'s......Page 399
5. Relationships among []'s......Page 401
References......Page 403
1. Introduction......Page 404
2. Notation and formulas......Page 407
3. Lemmas......Page 408
4. Proofs of theorems.......Page 409
1. Introduction......Page 413
2. Connection forms of a metric......Page 414
3. Difference tensors......Page 415
4. Subscalar pairs of metrics......Page 416
5. Metrics inducing the same volume element......Page 418
6. Isometries of convex hypersurfaces......Page 420
References......Page 429
1. Introduction......Page 430
2. Notation and formulas......Page 431
3. Lemmas......Page 434
4. Theorem......Page 436
References......Page 438
1. The signature theorem......Page 439
2. The G-signature and signature-defect......Page 441
References......Page 442
1. Introduction.......Page 443
2. Notation and formulas.......Page 446
3. Lemmas.......Page 447
4. Proofs of theorems.......Page 449
2. Some geometry of shells......Page 455
3. An inequality for the total absolute curvature......Page 463
4. Appendix......Page 467
References......Page 471
1. Pinched manifolds without boundary......Page 472
2. Doubling of a manifold......Page 473
3. Main theorem......Page 474
References......Page 476
1. Introduction......Page 477
3. Main theorem......Page 478
References......Page 481
0. Introduction......Page 482
1. Vector product and generalized covariant differentiation......Page 484
2. Curvatures and fundamental forms......Page 487
3. Infinitesimal conformal transformations and lemmas......Page 489
4. Integral formulas......Page 493
5. Congruence theorems......Page 496
1. Introduction......Page 498
3. Lemmas......Page 501
4. Proof of the theorems......Page 504
References......Page 507
1. Introduction.......Page 508
2. Curvature forms; Pontrjagin classes and numbers......Page 509
3. Doubling of a manifold......Page 510
4. Proof of theorem 1.2......Page 511
References......Page 512
1. Introduction......Page 513
2. Vector product and orthonormal frames......Page 514
3. Immersed submanifolds......Page 516
4. Integral formulas......Page 521
5. Characterizations of umbilical submanifolds......Page 524
References......Page 531
1. Introduction......Page 532
2. Vector product and orthonormal frames......Page 533
3. Immersed surfaces......Page 535
4. Proof of Theorem 1.1......Page 537
References......Page 540
1. INTRODUCTION......Page 541
2. NOTATION AND FORMULAS......Page 542
3. PROOF OF THEOREM 1......Page 545
4. PROOF OF THEOREM 2......Page 546
1. INTRODUCTION......Page 549
2. IMMERSED SUBMANIFOLDS IN EUCLIDEAN SPACE......Page 552
3. INTEGRAL FORMULAS FOR A PAIR OF IMMERSED MANIFOLDS......Page 554
4. PROOFS OF THEOREMS I AND II......Page 560
5. EXAMPLE......Page 562
Introduction......Page 566
1. Notation and Riemannian metrics.......Page 567
2. Complex structures.......Page 569
3. Almost complex structures.......Page 571
4. Complex operators.......Page 572
5. Calculations of d for 0- and 1-forms.......Page 573
6. Calculations of d for 2-forms.......Page 575
7. Proof of the theorem.......Page 580
8. Proofs of the corollaries.......Page 581
References......Page 582
Introduction......Page 583
1. Curvature operators.......Page 584
2. Euler-Poincare characteristic and curvature operators.......Page 588
3. Curvature operators of the form Rp = cAp on a Kahler manifold.......Page 589
4. Constancy of holomorphic sectional curvature.......Page 592
5. Constant zero holomorphic sectional curvature.......Page 595
References......Page 597
1. Introduction.......Page 599
2. Proof of Theorem 4.......Page 601
References......Page 607
1. Introduction.......Page 608
3. Almost complex structures.......Page 609
4. Almost L structures.......Page 611
5. Sectional curvatures.......Page 612
S 6. Kaehlerian structures.......Page 616
References......Page 619
1. Introduction.......Page 620
2. Riemannian structures.......Page 622
3. Almost complex structures.......Page 623
4. Almost C-structures.......Page 625
5. Holomorphic sectional curvature.......Page 628
6. Complex structures.......Page 635
References......Page 636
1. Introduction.......Page 637
2. Preliminaries.......Page 640
3. Almost L manifolds.......Page 642
4. Proof of the Theorem......Page 645
References......Page 649
S1. Introduction......Page 650
S2. Riemannian structures......Page 652
S3. Almost complex structures......Page 654
S4. General sectional curvatures......Page 656
S5. Holomorphic sectional curvatures......Page 657
S6. Holomorphic bisectional curvatures......Page 663
S7. The relationship among the three types of sectional curvatures......Page 666
References......Page 671
1. Introduction......Page 672
2. Almost complex structures......Page 673
3. Almost Hermitian structures......Page 675
4. Spectra of Riemannian manifolds......Page 677
5. AH24 manifolds......Page 679
6. The main theorem......Page 683
References......Page 687
Some Conditions for a Complex Structure......Page 688
References......Page 690
1. Introduction......Page 691
2. Almost Complex Structures......Page 692
3. Proofs......Page 693
References......Page 694
Remarks on Some Selected Papers......Page 697
Curriculum Vitae......Page 701
Bibliography of the Publications of Chuan-Chih Hsiung......Page 703
List of Ph.D. Theses Written Under the Supervision of Chuan-Chih Hsiung......Page 711
Permission......Page 713