文档介绍：AN INTRODUCTION TO
DIFFERENTIAL GEOMETRY WITH
APPLICATIONS TO ELASTICITY
Philippe G. Ciarlet
City University of Hong Kong
Contents
Preface 5
1 Three-dimensional differential geometry 9
Introduction.............................. 9
Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . 11
............................. 13
Volumes, areas, and lengths in curvilinear coordinates . . . . . . 16
................ 19
Necessary conditions satisfied by the metric tensor; the Riemann
curvaturetensor........................... 24
ExistenceofanimmersiondefinedonanopensetinR3 with a
prescribedmetrictensor....................... 25
Uniqueness up to isometries of immersions with the same metric
tensor................................. 36
Continuity of an immersion as a function of its metric tensor . . 44
2 Differential geometry of surfaces 59
Introduction.............................. 59
Curvilinear coordinates on a surface . . . . . . . . . . . . . . . . 61
First fundamental form . . . . . . . . . . . . . . . . . . . . . . . 65
Areas and lengths on a surface . . . . . . . . . . . . . . . . . . . 67
Second fundamental form; curvature on a surface . . . . . . . . . 69
Principal curvatures; Gaussian curvature . . . . . . . . . . . . . . 73
Covariant derivatives of a vector field defined on a surface; the
Gauß and Weingarten formulas . . . . . . . . . . . . . . . . . . . 79
Necessary conditions satisfied by the first and second fundamen-
tal forms: the Gauß and Codazzi-Mainardi equations; Gauß’
TheoremaEgregium......................... 82
Existence of a surface with prescribed first and second fundamen-
talforms................................ 85
Uniqueness up to proper isometries of surfaces with the same
fundamental forms . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Continuity of a surface