The geometry of total curvature on complete open surfaces

Shioya T., Shiohama K.0511067593, 0521450543

This independent account of modern ideas in differential geometry shows how they can be used to understand and extend classical results in integral geometry. The authors explore the influence of total curvature on the metric structure of complete, non-compact Riemannian 2-manifolds, although their work can be extended to more general spaces. Each chapter features open problems, making the volume a suitable learning aid for graduate students and non-specialists who seek an introduction to this modern area of differential geometry.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 9
1.1 The Riemannian metric……Page 13
1.2 Geodesics……Page 15
1.3 The Riemannian curvature tensor……Page 20
1.4 The second fundamental form……Page 26
1.5 The second variation formula and Jacobi fields……Page 29
1.6 Index form……Page 36
1.7 Complete Riemannian manifolds……Page 40
1.8 The short-cut principle……Page 43
1.9 The Gauss–Bonnet theorem……Page 46
2.1 The total curvature of complete open surfaces……Page 53
2.2 The classical theorems of Cohn-Vossen and Huber……Page 58
2.3 Special properties of geodesics on Riemannian planes……Page 67
3.1 The curvature at infinity……Page 87
3.2 Parallelism and pseudo-distance between curves……Page 90
3.3 Riemannian half-cylinders and their universal coverings……Page 103
3.4 The ideal boundary and its topological structure……Page 106
3.5 The structure of the Tits metric………Page 112
3.6 Triangle comparison……Page 117
3.7 Convergence to the limit cone……Page 123
3.8 The behavior of Busemann functions……Page 135
4.1 Preliminaries……Page 145
4.2 The topological structure of a cut locus……Page 152
4.3 Absolute continuity of the distance function of the cut locus……Page 161
4.4 The structure of geodesic circles……Page 168
5.1 The structures of S(C, t) and the cut locus of C……Page 177
5.2 The case where M is finitely connected……Page 181
5.3 The case where M is infinitely connected……Page 187
6.1 Preliminaries; the mass of rays emanating from a fixed point……Page 199
6.2 Asymptotic behavior of the mass of rays……Page 207
7.1 Properties of geodesics……Page 219
7.2 Jacobi fields……Page 230
7.3 The cut loci of a von Mangoldt surface……Page 242
8.1 The shape of plane curves……Page 255
8.2 Main theorems and examples……Page 260
8.3 The semi-regularity of geodesics……Page 264
8.4 Almost-regularity of geodesics; estimate of index……Page 274
8.5 The rotation numbers of proper complete geodesics……Page 278
8.6 The existence of complete geodesics arbitrarily close to infinity……Page 282
References……Page 287
Index……Page 293