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 /Lie bracket vanishing and coordinate systems/ Cartan's equations: connection and curvature forms, Bianchi identities

3/23 F  parallel transport around closed loops/ contracted Bianchi identity (divergence of scalar curvature)

3/28 W isometric immersions: Weingarten map, mean and Gauss curvatures, Gauss theorem, Coddazzi identity (for hypersurfaces in R^n)
Problem set 5 (from do Carmo): p. 106, no. 9; p.108 no. 10 (for part (a), use the fact the gradient of scalar is twice the divergence of (1,1) Ricci);
p. 141 11 c), d)

4/4 W curvature tensor of hyp space, sphere/spaces of constant curvature/irreducible components

4/6 F lecture canceled

4/11 W Conformal change of metric/ locally conformally flat manifolds

4/13 F Jacobi fields and applications/ conjugate points, cut locus
Conformally related metrics

4/18 W Second variation of length/ Index form, Jacobi's theorem, Jacobi's criterion

4/20 F Theorems of Hadamard and Bonnet-Myers/ Gauss-Bonnet theorem (start.)