## Differential GeometryThis 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. |

### Contents

Operations with Vectors | 1 |

Scalar triple product | 7 |

Space Curves | 12 |

The Jordan theorem as a problem in differential geometry in | 34 |

Additional properties of Jordan curves | 43 |

Four vertex theorem | 51 |

The Basic Elements of Surface Theory | 74 |

Torsion of asymptotic lines | 93 |

Geodetically convex domains | 231 |

The GaussBonnet formula applied to closed surfaces | 237 |

Vector fields on surfaces and their singularities | 239 |

Poincarés 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 | 259 |

Characterization of the sphere as a locus of umbilical points | 99 |

Analogues of polar coordinates on a surface | 105 |

Some Special Surfaces | 109 |

Developable surfaces in the large | 125 |

Developables as envelopes of planes | 131 |

Uniqueness of a surface for given gu and | 139 |

Inner Differential Geometry in the Small from the Extrinsic | 151 |

Parallel transport of vectors along curves in the sense of LeviCivita | 157 |

the geodetic curvature | 163 |

A general necessary condition for a shortest | 171 |

Geodesics as shortest arcs in the small | 178 |

Parallel fields from a new point of view | 184 |

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 |

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 |

Theory of Relativity 11 Special theory of relativity | 318 |

Relativistic dynamics | 323 |

The general theory of relativity | 326 |

The Wedge Product and the Exterior Derivative of Differential | 335 |

Vector differential forms and surface theory | 342 |

Scalar and vector products of vector forms on surfaces and their | 349 |

Minimal surfaces | 356 |

Appendix A Tensor Algebra in Affime Euclidean and Minkowski Spaces | 371 |

Appendix B Differential Equations | 388 |

396 | |

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### Common terms and phrases

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