MATH 568- RIEMANNIAN GEOMETRY I-SPRING 2018
Topics and bibliography
1/10 W No lecture (due to JMM 2018)
1/12 F No lecture (JMM 2018)
1/17 W Surfaces in R^n/ Definition of manifold/ Ex: discontinuous group actions
1/19 F Paracompactness, partitions of unity
1/24 W Tangent vectors, tangent bundle, differential of a smooth map
1/26 F Vector fields, one-forms, integral curves, Lie bracket
Problem set 1 (due Feb. 2)
1/31 W Interpretation of Lie bracket/ one-forms, two-forms, exterior derivative.
2/2 F Immersions, submanifolds, submersions./ Connections: examples in R^n and for surfaces
Problem set 2
(due W 2/14)
2/7 W Vector bundles and connections on them: existence. Riemannian metrics (existence), metric/compatible connections.
2/9 F Recovering connection from parallel transport/torsion-free
connections, Levi-Civita connection. Example of submanifolds.
2/14 W Geodesics, First variation formula for length
2/16 F Geodesic flow: local existence, homogeneity lemma. Exponential
map, normal balls. Examples: spheres, surfaces of revolution (Clairaut inv)
2/21 W Geodesics of hyperbolic plane: variational derivation, constants of motion, interpretation/ Gauss lemma (start)
Problem set 3 (due F 3/2)--from Do Carmo: Chapter 3, problems 1d, 5, 6, 7.
2/23 F Gauss lemma (conclusion)/ Geodesics minimize locally/ Second fundamental form
2/28 W Second ff and parallel transport/ tot. geod and tot umbillic submfds/ L-C connection of skew products
Geodesic equations in simple examples
3/2 F Geodesics of a skew product/ TG submanifolds of space forms/ Completeness: Hopf-Rinow theorem
3/7 W Geodesic connectivity; complete manifolds/ volume form, gradient and divergence--coordinate expressions
Problem set 4 (due F 3/23)--from do Carmo: p.104 no. 4/ p. 141 no. 11/p. 152 no 2, 5, 6
3/9 Hessian and Laplace-Beltrami operator/ induced volume on the boundary, divergence theorem/
curvature tensor: vanishing equivalent to local flatness
3/14, 3/16: SPRING BREAK
3/21 W parallel transport around closed loops/ Lie bracket vanishing and coordinate systems
3/23 F Differential Bianchi identity and diffeomorphism invariance/ Lorentzian non-completeness example