Geometry from a differentiable viewpoint

Cover......Page __sk_0000.djvu
Copyright......Page __sk_0002.djvu
Contents......Page __sk_0005.djvu
Introduction......Page __sk_0007.djvu
Part A. Prelude and themes: Synthetic methods and results......Page __sk_0011.djvu
1. Spherical geometry......Page __sk_0013.djvu
2. Euclid......Page __sk_0020.djvu
Euclid\'s theory of parallels......Page __sk_0026.djvu
Appendix. The Elements: Book I......Page __sk_0031.djvu
Uniqueness of parallels......Page __sk_0034.djvu
Equidistance and boundedness of parallels......Page __sk_0036.djvu
On the angle sum of a triangle......Page __sk_0038.djvu
Similarity of triangles......Page __sk_0041.djvu
The work of Saccheri......Page __sk_0044.djvu
The work of Gauss, Bolyai, and Lobachevskii......Page __sk_0049.djvu
5. Non-Euclidean geometry II......Page __sk_0055.djvu
The circumference of a circle......Page __sk_0066.djvu
Part B. Development: Differential geometry......Page __sk_0071.djvu
6. Curves......Page __sk_0073.djvu
Early work on plane curves (Huygens, Leibniz, Newton, Euler)......Page __sk_0076.djvu
The tractrix......Page __sk_0079.djvu
Directed curvature......Page __sk_0080.djvu
Digression: Involutes and evolutes......Page __sk_0082.djvu
7. Curves in space......Page __sk_0090.djvu
Appendix: On Euclidean rigid motions......Page __sk_0099.djvu
8. Surfaces......Page __sk_0105.djvu
The tangent plane......Page __sk_0111.djvu
The first fundamental form......Page __sk_0116.djvu
Area......Page __sk_0122.djvu
8 bis. Map projections......Page __sk_0126.djvu
Stereographic projection......Page __sk_0130.djvu
Central projection......Page __sk_0133.djvu
Mercator projection......Page __sk_0134.djvu
Azimuthal projection......Page __sk_0136.djvu
Sample map projections......Page __sk_0137.djvu
Euler\'s work on surfaces......Page __sk_0141.djvu
The Gauss map......Page __sk_0144.djvu
10. Metric equivalence of surfaces......Page __sk_0155.djvu
Special coordinates......Page __sk_0161.djvu
11. Geodesics......Page __sk_0167.djvu
Euclid revisited I: The Hopf-Rinow theorem......Page __sk_0175.djvu
12. The Gauss-Bonnet theorem......Page __sk_0181.djvu
Euclid revisited II: Uniqueness of lines......Page __sk_0185.djvu
Compact surfaces......Page __sk_0186.djvu
A digression on curves......Page __sk_0190.djvu
13. Constant-curvature surfaces......Page __sk_0196.djvu
Euclid revisited III: Congruences......Page __sk_0201.djvu
The work of Minding......Page __sk_0202.djvu
Part C. Recapitulation and coda......Page __sk_0209.djvu
14. Abstract surfaces......Page __sk_0211.djvu
Hilbert\'s theorem......Page __sk_0213.djvu
Abstract surfaces......Page __sk_0216.djvu
15. Modeling the non-Euclidean plane......Page __sk_0227.djvu
The Beltrami disk......Page __sk_0230.djvu
The Poincaré disk......Page __sk_0234.djvu
The Poincaré half-plane......Page __sk_0237.djvu
16. Epilog: Where from here?......Page __sk_0252.djvu
Manifolds (differential topology)......Page __sk_0253.djvu
Vector and tensor fields......Page __sk_0257.djvu
Metrical relations (Riemannian manifolds)......Page __sk_0259.djvu
Curvature......Page __sk_0262.djvu
Covariant differentiation......Page __sk_0271.djvu
Riemann\'s Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry......Page __sk_0279.djvu
Appendix: Notes on selected exercises......Page __sk_0289.djvu
Bibliography......Page __sk_0307.djvu
Symbol index......Page __sk_0313.djvu
Name index......Page __sk_0314.djvu
Subject index......Page __sk_0315.djvu