# Differential Geometry

John Wiley & Sons, Jan 18, 1989 - Mathematics - 404 pages
This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### Contents

 Operations with Vectors 1 The vector notation 1 Addition of vectors 2 Multiplication by scalars 3 Vector product 5 Scalar triple product 6 Invariance under orthogonal transformations 7 Vector calculus 11 Plane Carves 1 Introduction 12
 The Gauss equations 134 The Christoffel symbols evaluated 135 The Weingarten equations 136 Uniqueness of a surface for given gk and Lk 138 The theorema egregium of Gauss 139 How Gauss may have hit upon his theorem 141 Compatibility conditions in general 143 CodazziMainardi equations 144

 Change of parameters 14 Invariance under changes of parameter 16 Orientation of a curve 18 Length of a curve 19 Arc length as an invariant 20 Curvature of plane curves 21 The normal vector and the sign of r 23 Formulas for c 26 Existence of a plane curve for given curvature k 27 Frenet equations for plane curves 28 Evolute and involute of a plane curve 29 Envelopes of families of curves 31 The Jordan theorem as a problem in differential geometry in the large 34 Additional properties of Jordan curves 41 The total curvature of a regular Jordan curve 45 Simple closed curves with 0 as boundaries of convex point sets 46 Four vertex theorem 48 Chapter in Space Carres 1 Regular curves 53 Length of a curve 54 Principal normal and osculating plane 55 Binomial vector 57 The Frenet equations for space curves 58 The Darboux vector 62 Formulas for and r 63 Canonical representation of a curve 64 Existence and uniqueness of a space curve for given s tj 65 What about 0? 67 Another way to define space curves 68 Some special curves 70 The Basic Elements of Surface Theory 1 Regular surfaces in Euclidean space 74 Change of parameters 75 Curvilinear coordinate curves on a surface 76 Tangent plane and normal vector 77 Length of curves and first fundamental form 78 Angle measurement on surfaces 80 Area of a surface 82 A few examples 83 Second fundamental form of a surface 85 Osculating paraboloid 86 Curvature of curves on a surface 88 Principal directions and principal curvatures 91 Mean curvature H and Gaussian curvature K 92 Another definition of the Gaussian curvature K 93 Lines of curvature 95 Third fundamental form 98 Characterization of the sphere as a locus of umbilical points 99 Asymptotic lines 100 Introduction of special parameter curves 101 Asymptotic lines and lines of curvature as parameter curves 103 Analogues of polar coordinates on a surface 104 Some Special Surfaces 1 Surfaces of revolution 109 Developable surfaces in the small made up of parabolic points 114 Edge of regression of a developable 118 Why the name developable? 120 Developable surfaces in the large 121 Developables as envelopes of planes 129 The Partial Differential Equations of Surface Theory 1 Introduction 133
 Existence of a surface with given gu and Llk 146 An application of the general theory to a problem in the large 148 Inner Differential Geometry in the Small from the Extrinsic Point of View 1 Introduction Motivations for the basic concepts 151 Approximate local parallelism of vectors in a surface 155 Parallel transport of vectors along curves in the sense of LeviChrita 157 Properties of parallel fields of vectors along curves 160 Parallel transport is independent of the path only for surfaces hav ing K m 0 162 the geodetic curvature 163 lines with 0 165 Geodetic lines as candidates for shortest arcs 167 Straight lines as shortest arcs in the Euclidean plane 168 A general necessary condition for a shortest arc 171 Geodesies in the small and geodetic coordinate systems 174 Geodesies as shortest arcs in the small 178 Further developments relating to geodetic coordinate systems 179 Surfaces of constant Gaussian curvature 183 Parallel fields from a new point of view 184 Models provided by differential geometry for nonEuclidean geometries 185 Parallel transport of a vector around a simple closed curve 191 Derivation of the GaussBonnet formula 195 Consequences of the GaussBonnet formula 196 Tchebychef nets 198 Differential Geometry in the Large 1 Introduction Definition of ndimensional manifolds 203 Definition of a Riemannian manifold 206 Facts from topology relating to twodimensional manifolds 211 Surfaces in threedimensional space 217 Abstract surfaces as metric spaces 218 Complete surfaces and the existence of shortest arcs 220 Angle comparison theorems for geodetic triangles 227 Geodetically convex domains 231 The GaussBonnet formula applied to closed surfaces 237 Vector fields on surfaces and their singularities 239 Poincares theorem on the sum of the indices on closed surfaces 244 Conjugate points Jacobis conditions for shortest arcs 247 The theorem of BonnetHopfRinow 254 Synges theorem in two dimensions 255 Covering surfaces of complete surfaces having K 0 261 Hilberts theorem on surfaces in E3 with K 1 265 The form of complete surfaces of positive curvature in threedimen sional space 272 Intrinsic Differential Geometry of Manifolds Relativity 1 Introduction 282 Tensor Calculus in Affine and Euclidean Spaces 2 Affine geometry in curvilinear coordinates 284 Tensor calculus in Euclidean spaces 287 Tensor calculus in mechanics and physics 292 Tensor Calculus and Differential Geometry in General Manifolds 5 Tensors in a Riemannian space 294 Basic concepts of Riemannian geometry 296 Parallel displacement Necessary condition for Euclidean metrics 300 Normal coordinates Curvature in Riemannian geometry 307 Geodetic lines as shortest connections in the small 310 Geodetic lines as shortest connections in the large 311 Special theory of relativity 318 Relativistic dynamics 323 The general theory of relativity 326 The Wedge Product and die Exterior Demati?e of Differential 335 Appendix A Tensor Algebra in Affine Euclidean and Minkowski Spaces 371 Appendix B Differential Equations 388 Bibliography 396 Copyright