## 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. |

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### 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 |

396 | |

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### Common terms and phrases

angle arbitrary arc length assumed asymptotic lines basis vectors boundary points calculation Chapter closed surface coefficients compact components condition Consider const constant continuous convex convex set coordinate system coordinate vectors covariant curvilinear coordinates defined definition differential forms differential geometry discussion domain Euclidean space exists fact fixed follows formula Frenet equations functions fundamental form Gaussian curvature geodesies geodetic circle geodetic lines geodetic polar coordinates given hence holds integral introduced invariant isometric line element linear lines of curvature manifold mapping metric neighborhood normal vector notion one-to-one orientation orthogonal osculating pair of points parabolic points parallel field parallel transport parameter curves plane curves positive problem proof proved regular curve regular surface result Riemannian Section seen segment shown singularities space curve sphere straight line tangent plane tangent space tangent vector tensor theorem theory three-space tion topological triangle uniquely determined unit vector vanish vector field zero