MATH 567- RIEMANNIAN GEOMETRY- FALL 2008

SYLLABUS

8/21   Th       C^k surfaces in euclidean space/non-examples/local form of immersions and
of submersions/examples of surfaces: preimage of regular values, SL_n(R), orthogonal group
problems for lecture 1 (due 9/4)

8/26     Tu      Definition of C^k manifold/ non-Hausdorff example/ properly discontinuous actions and
quotients/ tangent space at a point (as equiv classes of triples (U,phi,v)).
problems for lecture 2 ( due 9/4)

8/28     Th      Tangent vectors as derivations/tangent bundle/immersions, embeddings, submanifolds/
Proper maps/ vector fields/ local flow of a vector fields/flows on compact mfd are global
problems for lecture 3 (due 9/4): problems 0.2 and 0.8 in text.  Also:  show that a proper map
(between manifolds, say) is closed.

9/2       Tu        Lie bracket: definition, Jacobi identity, invariance under immersions, geometric meaning.
Frobenius' theorem (statement)
problems for lecture 4 (due 9/11)

9/4        Th      compact manifolds embed in euclidean space/ partitions of unity: idea of proof/applications:
Riemannian metrics, differentiable Urysohn's lemma

9/9 ,9/11 Tu,Th      Whitney's immersion and embedding theorems  (draft; includes 4 homework problems, due 9/18)

9/16                   affine connections/ Levi-Civita connection
problems from Ch.2 : 2,3,8 (due 9/25)  (I also added a problem to the end of the Whitney handout)

9/18                    induced connection, parallel transport on the sphere, Hessian

9/23                     tensors: covariant derivative, Lie derivative. Geodesics; the geodesic flow, geodesics of tne n-sphere.
Problems from Ch 3: 1,2 (due 10/2))

9/25                     geodesics: exponential map, the Gauss lemma, isometries of hyp. plane

9/30                      geodesics: normal and totally normal neighborhoods, local minimization, geods. of hyp. plane
Probems from Ch. 3: 5, 6 (Killing fields- due 10/2)

10/2                    Riemannian distance. Completeness: definition, examples of complete manifolds. Hopf-Rinow,
geodesic connectedness of complete manifolds

10/7                   Homework on complete manifolds (ch. 7)- these will be discussed in class starting 10/ 7, following the assignment:
Kilpatrick: 2,6 Orick: 3,7 Virk: 4,8 Bunn: 5,12 Remark: written solutions due 10/18.

10/9                     Fall break. Handout: Geodesics of a Lorentzian manifold
(5 pages, includes exercises. Will be discussed next week)

10/14                   Discussion of problems on completeness

10/16                   Curvature tensor, sectional curvature.
Problems (ch.  4 : 4, 6, 7, 8, 10)- turn in at least three by Thursday 10/23; they're all
interesting, and will be discussed in class.)

10/21                   Jacobi fields- Jacobi equation, slns in constant curvature, relationship with exponential map
Problems: (Ch.5: 1,6,7)- due Nov 4

10/23                 Conjugate points, Hadamard Theorem, Bianchi II

10/28                  Geometry of submanifolds: 2nd fundamental form, shape operator, example of graphs
Einstein tensor is divergence-free

10/30                    no class (talk at Columbia)

11/4                   Geometry of submanifolds: Gauss equation and applications

11/6                   Fundamental equations for submanifolds/ comments on hw problems
Hw problems, p.139: 1,2,5,8 (due 11/13)

11/11                The local gauss-Bonnet formula

11/13                Gauss-Bonnet formula, problems on surfaces

11/18                 First and second variations of arc length/ Bonnet-Meyrs theorem
Problems from ch. 9: 1,2,4,6 (due 11/25)

11/20                 The index theorem / cut locus

11/25