CARNEGIE INSTITUTE OP TECHNOLOGY
LIBRARY
PRESENTED BY Dr. Lloyd L, Dines
A TREATISE ON THE DIFFERENTIAL GEOMETRY OF...
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CARNEGIE INSTITUTE OP TECHNOLOGY
LIBRARY
PRESENTED BY Dr. Lloyd L, Dines
A TREATISE ON THE DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
BY
IJTTHEU PFAIILKR EISENHAIIT PKOPKHHOIt OK MATUKMATK'H IN 1'ltINCMTON UNIVJCIWITY
GOT AND COMPANY BOSTON
NEW YOKE
CHICAGO LONDON
COPYBIUHT,
liKH),
BY
LUTHER PKAHLKtt KlSKNUAttT
(JINN
ANU COMPANY
PKIKTORS
Bt)STON
I'KO U.S.A.
PREFACE This book is a development from courses which I have given in Princeton for a number of years. During this time I have come to feel that more would be accomplished by my students if they had an introductory treatise written in English and otherwise adapted to the use of men beginning their graduate work. Chapter I is devoted to the theory of twisted curves, the method in general being that which is usually followed in discussions of this subject. But in addition I have introduced the idea of moving axes, and have derived the formulas pertaining thereto from the previously obtained Freiiet-Serret fornmlas. familiar with a
volume
method which
is
In this
way
is made in Darboux by
the student
similar to that used
and to that of Cesaro in his Gcomctria not only of great advantage in the treatment of certain topics and in the solution of problems, but it is valuable iu developing geometrical thinking. The remainder of the book may be divided into threo parts. The iirst, consisting of Chapters II-VI, deals with the geometry of a surface in the neighborhood of a point and the developments therefrom, such as curves and systems of curves defined by differential equations. To a large extent the method is that of Gauss, by which the properties of a surface are derived from the discussion of two qxiadratie differential forms. However, little or no space is given to the algebraic treatment of differential forms and their invariants. In addition, the method of moving axes, as defined in the first chapter, has been extended so as to be applicable to an investigation of the properties of surf ac.es and groups of surfaces. The extent of the theory concerning ordinary points is so great that no attempt has been made to consider the exceptional problems. Por a discussion of uch questions as the existence of integrals of differential equations and boundary conditions the reader must consult the treatises which deal particularly with these subjects. lu Chapters VII and VIII the theory previously developed is applied to several groups of surfaces, such as the quadrics, ruled surfaces, minimal surfaces, surfaces of constant total curvature, and surfaces with plane and spherical lines of curvature*
the
tirst
Ittiriiiseca.
of his Lepons,
This method
is
iii
PREFACE
iv
The idea of applicability of surfaces is introduced in Chapter IIT as a particular case of conformal representation, and throughout the book attention is called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely with the recent method of Weingarten and its developments. The remaining four chapters are devoted to a discussion of infinitesimal
deformation of surfaces, congruences of straight Hues and of circles, triply orthogonal systems of surfaces. It will be noticed that the book contains many examples, and the
and
student will find that whereas certain of
them
are merely direct
the formulas, others constitute extensions of the theory which might properly be included as portions of a more extensive treatise. At first I felt constrained to give such references as would enable the reader to consult the journals and treatises from applications of
which some
problems were taken, but finally
it seemed best remark that the flncyklopadie der mathematisc7ien Wissensckaften may be of assistance. And the same
of these
to furnish, no such key, only to
be said about references to the sources of the subject-matter of Many important citations have been made, but there has not been an attempt to give every reference. However, I desire to acknowledge niy indebtedness to the treatises of Uarboux, Biancln, and Scheffers. But the difficulty is that for many years I have consulted these authors so freely that now it is impossible for me to say,
may
the book.
except in certain cases, what specific debts I owe to each. In its present form, the material of the first eight chapters has been given to beginning classes in each of the last two years ; and the remainder of the book, with certain enlargements, has constituted an advanced course which has been followed several times. It is impossible for me to give suitable credit for the suggestions made and the assistance rendered by students during these years, but I am
my
conscious of helpful suggestions
made by my
Veblen, Maclnnes, and Swift, and by
colleagues, Professors
former colleague, Professor Bliss of Chicago. I wish also to thank Mr. A. K. Xrause for making the drawings for the figures. It remains for me to express my appreciation of the courtesy
shown by Ginn and Company, and
my
of the assistance given
by them
during the printing of this book.
LUTHER PFAHLEK EISENHAKT
CONTENTS CHAPTER
I
CURVES IN SPACE PAGE
SECTION 1.
2.
3. 4. 5. C.
7. 8. 9.
10. 11.
PARAMETRIC EQUATIONS OF A CURVE OTHER FORMS OF THE EQUATIONS OF A CURVE LINEAR ELEMENT TANGENT TO A CURVE ORDER OF CONTACT. NORMAL PLANK CURVATURE. RADIUS OF FIRST CURVATURE OSCULATING PLANE PRINCIPAL NORMAL AND BINOUMAL ... OSCULATING CIKCLK. CENTER OF FIRST CURVATURE TORSION. FRENKT-SKRRET FORMULAS ... FORM OF CURVE IN THE NEIGHBORHOOD OK A POINT. THK SIN OF TORSION
....
.
12. 13. 14.
15.
17.
18. 19.
20. 21.
22.
4 6
8 9
10 12 14 16
18
CYLINDRICAL HELICES INTRINSIC EQUATIONS. FUNDAMENTAL THEOREM RICCATI EQUATIONS THE DETERMINATION OF THE COORDINATES OF A CURVE DEFINED BY ITS INTRINSIC EQUATIONS MOVING TRIHEDRAI ILLUSTRATIVE EXAMPLES OSCULATING SPHERE
20 22
BERTRAND CURVES TINGENT SURFACE OK A CURVE INVOLUTES AND EVOLUTKS OF A CURVE MINIMAL CURVES
39
.
.
.
.
.
...
16.
1
3
.
CHAPTER
.
25
27 30 33
37 41
43 47
II
CURVILINEAR COORDINATES ON A SURFACE. ENVELOPES 23. 24. 25.
26.
PARAMETRIC EQUATIONS OF A SURFACE PARAMETRIC CURVES TANGENT PLANE ONE-PARAMETER FAMILIES OF SURFACES. ENVELOPES
....
52 54 56 59
CONTENTS
vi
PAGE
SECTION 27.
28. 29.
DEVELOPABLE SURFACES. RECTIFYING DEVELOPABLE APPLICATIONS OF THE MOVING TRIHEDRAL ENVELOPE OF SPHERES. CANAL SURFACES
CHAPTER
....
61
64
66
III
LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAMETERS. CONFORMAL REPRESENTATION 30. 31. 32. 33. 34. 35.
36. 37.
38.
39. 40. 41.
42. 43. 44.
45. 46.
47.
LINEAR ELEMENT ISOTROPIC DEVELOPABLE TRANSFORMATION OF COORDINATES ANGLES BETWEEN CURVES. THE ELEMENT OF AREA ... FAMILIES OF CURVES MINIMAL CURVES ON A SURFACE VARIATION OF A FUNCTION DIFFERENTIAL PARAMETERS OF THE FIRST ORDER DIFFERENTIAL PARAMETERS OF THE SECOND ORDER SYMMETRIC COORDINATES ISOTHERMIC AND ISOMETRIC PAKAMETERS ISOTHERMIC ORTHOGONAL SYSTEMS CONFORMAL REPRESENTATION ISOMETRIC REPRESENTATION. APPLICABLE SURFACES CONFORMAL REPRESENTATION OF A SURFACE UPON ITSELF CONFORMAL REPRESENTATION OF THE PLANE SURFACES OF REVOLUTION CONFORMAL REPRESENTATIONS OF THE SPHERE .
.
.
.
.
.
....
.
.
.
.
.
.
70 72 72
74
78 81 82 84.
88 91
93
95 98
100 101
104
107 109
CHAPTEE IV GEOMETRY OF A SURFACE IN THE NEIGHBORHOOD OF A POINT
50.
FUNDAMENTAL COEFFICIENTS OF THE SECOND ORDER RADIUS OF NORMAL CURVATURE PRINCIPAL RADII OF NORMAL CURVATURE
51.
LINES OF CURVATURE.
48. 49.
52. 53. 54.
....
EQUATIONS OF RODRIGUES TOTAL AND MEAN CURVATURE EQUATION OF EULER. DUPIN INDICATRIX CONJUGATE DIRECTIONS AT A POINT. CONJUGATE SYSTEMS
114 117
118 121 123
124 .
126
130
57.
ASYMPTOTIC LINES. CHARACTERISTIC LINES CORRESPONDING SYSTEMS ON Two SURFACES GEODESIC CURVATURE. GEODESICS
58.
FUNDAMENTAL FORMULAS
133
55.
56.
128 131
CONTENTS
vii
SECTION 59. 60. 61.
62.
PAC E -
GEODESIC TORSION SPHERICAL REPRESENTATION RELATIONS BETWEEN A SURFACE AND ITS SPHERICAL REPRESENTATION HELICOIDS
137 141
.
143
146
CHAPTER V FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL 63.
CHRISTOFFKL SYMBOLS
64.
71.
THE EQUATIONS OF GAUSS AND OF CODAZZI FUNDAMENTAL THEOREM FUNDAMENTAL EQUATIONS IN ANOTHER FORM TANGENTIAL COORDINATES. MEAN EVOLUTE THE MOVING TRIHEDRAL FUNDAMENTAL EQUATIONS OF CONDITION LINEAR ELEMENT. LINES OF CURVATURE CONJUGATE DIRECTIONS AND ASYMPTOTIC DIRECTONS. SPHER-
72.
FUNDAMENTAL RELATIONS AND FORMULAS
174
73.
PARALLEL SURFACES SURFACES OF CENTER FUNDAMENTAL QUANTITIES FOR SURFACES OF CENTER SURFACES COMPLEMENTARY TO A GIVEN SURFACE
177
65. 66. 67. 68.
69.
70.
152
ICAL REPRESENTATION
74.
75. 76.
153 157
160 162 166
168 171
172
179 .
.
.
181
184
CHAPTER VI SYSTEMS OF CURVES. GEODESICS 77. 78.
79. 80. 81. 82.
83. 84.
85. 86. 87.
88.
89. 90.
ASYMPTOTIC LINES SPHERICAL REPRESENTATION OF ASYMPTOTIC LINES FORMULAS OF LELIEUVRE. TANGENTIAL EQUATIONS CONJUGATE SYSTEMS OF PARAMETRIC LINES. INVERSIONS SURFACES OF TRANSLATION ISOTHERMAL-CONJUGATE SYSTEMS SPHERICAL REPRESENTATION OF CONJUGATE SYSTEMS TANGENTIAL COORDINATES. PROJECTIVE TRANSFORMATIONS EQUATIONS OF GEODESIC LINES GEODESIC PARALLELS. GEODESIC PARAMETERS GEODESIC POLAR COORDINATES AREA OF A GEODESIC TRIANGLE LINES OF SHORTEST LENGTH. GEODESIC CURVATURE GEODESIC ELLIPSES AND HYPERBOLAS
189
.... ....
.
.
.
.
.
.
....
191
193
195 197 198 200 201
204 206
207 209
212 213
CONTENTS
viii
PAGE
SECTION-
01. Ot2.
D3.
94,
SURFACES OF LIOUVILLE INTEGRATION OF THE EQUATION or GEODESIC LINKS GEODESICS ON SURFACES OF LIOUVILLE LINES OF SHORTEST LENGTH. ENVELOPE OF G-EODEMCS
CHAPTEE
211 .
.
.
215
.
.
218
.
.
220
VII
QUADRICS. RULED SURFACES. MINIMAL SURFACES 05. 96.
97. 98.
99.
100.
101. 102. 103. 101.
105. 106.
CONFOCAL QUADRICS. ELLIPTIC COORDINATES FUNDAMENTAL QUANTITIES FOR CENTRAL QUAPUICS FUNDAMENTAL QUANTITIES FOR THE PARABOLOIDS LINES OF CURVATURE AND ASYMPTOTIC LINES ON QUADRICS GEODESICS ox QUADRICS GEODICSICS THROUGH THE UMBILICAL POINTS ELLIPSOID REFERRED TO A POLAR GEODESIC SYSTEM .
.
229
.
.
.
230
.
232
....
.
PROPERTIES OF QUADRICS EQUATIONS OF A RULED SURFACE LINE OF STRICTION. DEVELOPABLE SURFACES CENTRAL PLANE. PARAMETER OF DISTRIBUTION* PARTICULAR FORM OF THE LINEAR ELEMENT .
226
.
251 .
230
.
.
236
.
.
239
...
211
.
.
.
242
.
.
.
211
.
.
.
247
248
113.
ASYMPTOTIC LINES. ORTHOGONAL PARAMETRIC SYSTEMS MINIMAL SURFACES LINES OF CURVATURE AND ASYMPTOTIC LINES. ADJOINT MINIMAL SURFACES MINIMAL CURVES ox A MINIMAL SURFACE DOUBLE MINIMAL SURFACES ALGEBRAIC MINIMAL SURFACES ASSOCIATE SURFACES
lit.
FORMULAS OF SCHWARZ
264
107. 108.
100.
.
.
....
...
110.
111. 112.
.
....
CHAPTEE
....
250
253
.
.
254
.
.
258
.
260 263
VIII
SURFACES OF CONSTANT TOTAL CURVATURE. TV-SURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CUR-
VATURE
110.
SPHERICAL SURFACES OF REVOLUTION PSEUDOSPHERICAL SURFACES OF REVOLUTION GEODESIC PARAMETRIC SYSTEMS. APPLICABILITY TRANSFORMATION OF HAZZIDAKIS TRANSFORMATION OP BIANCHI
120.
TRANSFORMATION OF BACKLUND
121.
THEOREM OF PERM UT ABILITY
115. 116. 117.
118.
270 272 275
278 280 .
284 286
CONTENTS
ix
SECTION
PAGE
122.
TRANSFORMATION OF LIE
123.
IK-SURFACES.
124.
EVOLUTE OP A TF-SURFACE SURFACES OF CONSTANT MEAN CURVATURE RULED TK-SuRPACEs SPHERICAL REPRESENTATION OF SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN ONE SYSTEM.
125. 126. 127.
128. 129.
2S9
FUNDAMENTAL QUANTITIES
2,01 .
.
.
.
.
SURFACES OF MONGE 130. 131.
132. 133. 134.
294 296 299
300 302
305
MOLDING SURFACES SURFACES OF JOACHIMSTHAL SURFACES WITH CIRCULAR LINES OF CURVATURE CYCLIDES OF DUPIN SURFACES WITH SPHERICAL LINES OF CURVATURE
.
307
.
.
308
310
312 IN
ONE
SYSTEM
314
CHAPTER IX DEFORMATION OF SURFACES 135.
136. 137.
138. 139.
PROBLEM OF MINDING. SURFACES OF CONSTANT CURVATURE SOLUTION OF THE PROBLEM OF MINDING DEFORMATION OF MINIMAL SURFACES SECOND GENERAL PROBLEM OF DEFORMATION DEFORMATIONS WHICH CHANGE A CURVE ON THE SURFACE INTO A GIVEN CURVE IN SPACE LINES OF CURVATURE IN CORRESPONDENCE CONJUGATE SYSTEMS IN CORRESPONDENCE ASYMPTOTIC LINES IN CORRESPONDENCE. DEFORMATION OF A RULED SURFACE METHOD OF MINDING PARTICULAR DEFORMATIONS OF RULED SURFACES
....
.
.
140.
141. 142.
143.
144.
.
321
.
323
.... ....
.
327 331
333 336 338
342 344 345
CHAPTER X DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN
118.
REDUCED FORM OF THE LINEAR ELEMENT GENERAL FORMULAS THE THEOREM OF WEINGARTEN OTHER FORMS OF THE THEOREM OF WEINGARTEN
149.
SURFACES APPLICABLE TO A SURFACE OF REVOLUTION
145. 146.
147.
351
.
.
353 355
.... .
.
357
362
CONTENTS
x
PAGE
SEi'Tiox
150. 151.
-
364
.
366
MINIMAL LINES ox TUE SPHERE PARAMETRIC SURFACES OF GOURSAT. SURFACED APPLICABLE TO CERTAIN PARABOLOIDS ... .
.
.
CHAPTER XI INFINITESIMAL DEFORMATION OF SURFACES 152.
GENERAL PROBLEM
373
CHARACTERISTIC FUNCTION 154. ASYMPTOTIC LINES PARAMETRIC 155. ASSOCIATE SURFACES
153.
156. 157. 158. 159.
374
.
.
376
.
.
378
PARTICULAR PARAMETRIC CURVES Sv SQ RELATIONS BETWEEN THREE SURFACES SURFACES RESULTING FROM AN INFINITESIMAL DEFORMATION ISOTHERMIC SURFACES
CHAPTER
379 382
385 387
XII
RECTILINEAR CONGRUENCES 160. DEFINITION OF
A CONGRUENCE.
SPHERICAL REPRESENTATION
392
161.
NORMAL CONGRUENCES. RULED SURFACES OF A CONGRUENCE
393
162.
LIMIT POINTS. PRINCIPAL SURFACES
163.
DEVELOPABLE SURFACES OF A CONGRUENCE. FOCAL SURFACES ASSOCIATE NORMAL CONGRUENCES DERIVED CONGRUENCES FUNDAMENTAL EQUATIONS OF CONDITION SPHERICAL REPRESENTATION OF PRINCIPAL SURFACES AND OF DEVELOPABLES FUNDAMENTAL QUANTITIES FOR THE FOCAL SURFACES ISOTROPIC CONGRUENCES CONGRUENCES OF GUICHARD PSEUDOSPHERICAL CONGRUENCES
415
TF-CONGRUENCES CONGRUENCES OF RIBAUCOUR
417 420
164. 165.
166. 167.
168. 169.
170. 171. 172. 173.
...
395
.
CHAPTER
.
.
398
401
403 406
407 409 412 414
XIII
CYCLIC SYSTEMS GENERAL EQUATIONS OF CYCLIC SYSTEMS CYCLIC CONGRUENCES 176. SPHERICAL REPRESENTATION OF CYCLIC CONGRUENCES 174.
426
175.
431 .
.
.
432
CONTENTS
xi
PAGE
SECTION 177. 178.
179.
180.
SURFACES ORTHOGONAL TO A CYCLIC SYSTEM NORMAL CYCLIC CONGRUENCES CYCLIC SYSTEMS FOR WHICH THE ENVELOPE OF THE PLANES OF THE CIRCLES is A CURVE CYCLIC SYSTEMS FOR WHICH THE PLANES OF THE CIRCLES PASS THROUGH
A POINT
436 437 439
440
CHAPTEE XIV TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 181.
TRIPLE SYSTEM OF SURFACES ASSOCIATED WITH A CYCLIC
182.
GENERAL EQUATIONS. THEOREM OF DUPIN
183.
EQUATIONS OF LAME TRIPLE SYSTEMS CONTAINING ONE FAMILY OF SURFACES OF REVOLUTION TRIPLE SYSTEMS OF BIANCHI AND OF WEINGARTEN THEOREM OF RIBAUCOUR
SYSTEM
184.
185.
186. 187.
188.
....
THEOREMS OF DARBOUX TRANSFORMATION OF COMBESCURE
INDEX
.
...
446 447 449 451 452 457 458 461
467
DIFFERENTIAL GEOMETRY CHAPTER
I
CURVES IN SPACE 1. Parametric equations of a curve. Consider space referred to fixed rectangular axes, and let (.r, y, z) denote as usual the coordinates of a point with respect to these axes. In the plane z draw a circle of radius r and center (a, b). The coordinates of a
=
P
point
on the
circle
can be expressed in the form
x = a + r cos u,
(1)
y
= b + rsmu,
z
= 0, P makes
where u denotes the angle which the radius to with the As u varies from to 360, the point P describes the circle. The quantities a, 5, r determine the position and size of the circle, whereas u determines the position of a point upon it. In this sense it is a variable or parameter for the positive a-axis.
And
circle.
equations
(1) are called
parametric
equations of the circle.
A
straight line in space
point on a,
Let
,
7.
P
it,
The
Jg(a, 5,
