Buch, Englisch, 322 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1070 g
Reihe: Universitext
Buch, Englisch, 322 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1070 g
Reihe: Universitext
ISBN: 978-3-540-20493-0
Verlag: Springer Berlin Heidelberg
This book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
1 Differential manifolds.- 1.A From submanifolds to abstract manifolds.- 1.B The tangent bundle.- 1.C Vector fields.- 1.D Baby Lie groups.- 1.E Covering maps and fibrations.- 1.F Tensors.- 1.G. Differential forms.- 1.H Partitions of unity.- 2 Riemannian metrics.- 2.A Existence theorems and first examples.- 2.B Covariant derivative.- 2.C Geodesies.- 2.D A glance at pseudo-Riemannian manifolds.- 3 Curvature.- 3.A. The curvature tensor.- 3.B. First and second variation.- 3.C. Jacobi vector fields.- 3.D. Riemannian submersions and curvature.- 3.E. The behavior of length and energy in the neighborhood of a geodesic.- 3.F Manifolds with constant sectional curvature.- 3.G Topology and curvature: two basic results.- 3.H. Curvature and volume.- 3.I. Curvature and growth of the fundamental group.- 3.J. Curvature and topology: some important results.- 3.K. Curvature tensors and representations of the orthogonal group.- 3.L. Hyperbolic geometry.- 3.M. Conformai geometry.- 4 Analysis on manifolds.-4.A. Manifolds with boundary.- 4.B. Bishop inequality.- 4.C. Differential forms and cohomology.- 4.D. Basic spectral geometry.- 4.E. Some examples of spectra.- 4.F The minimax principle.- 4.G Eigenvalues estimates.- 4.H. Paul Levy’s isoperimetric inequality.- 5 Riemannian submanifolds.- 5.A. Curvature of submanifolds.- 5.B Curvature and convexity.- 5.C Minimal surfaces.- A Some extra problems.- B Solutions of exercises.- List of figures.
