Lecturer: Detlef Gromoll, Math 5-116
Phone: 632-8290, email: firstname.lastname@example.org
Classes: MW 3:20-4:40pm, Harriman 115
Office Hours: By appointment
About this course: We will conclude our coverage of basic classical and current differential geometry in last year's MAT 568/569 sequence, but at a more advanced topics oriented level. Familiarity with much of the material presented in those courses will be needed, including some elementary Lie Group Theory. It will be possible to fill in gaps during the semester. The entry point for MAT 644 is approximately the book by Cheeger and Ebin; but no particular text will be used exclusively this Fall - we rather provide diverse references whenever needed.
Topics: Global riemannian geometry - review of the geometry of geodesics, second variation techniques; metric structure of a riemannian space, completeness and the Hopf-Rinow Theorem; basic comparison theory for distances, volumes, and angles - the Theorems of Rauch, Bishop-Gromov, and Toponogov; convexity; curvature and topology - structure results for spaces of nonpositive and nonnegative sectional curvature, Hadamard-Cartan Theorem, sphere and soul theorems; finiteness theorems; some results for ricci and scalar curvature; the quest for examples of nonnegatively curved spaces - homogeneous and symmetric spaces, riemannian submersions; new strings of examples (touching on earlier work of Berger, Wallach, Eschenburg, Shah-Yang, Wei, and others, as well as on the remarkable recent results of Grove-Ziller and Wilking).
What was covered in MAT 568/569 last year?
Review of smooth manifolds, tangent bundle, vector fields, Lie derivatives, Frobenius Theorem, Lie groups, homogeneous spaces; tensor fields and covariant derivatives on vector bundles, Koszul calculus along maps, parallel transport and curvature, geodesics and exponential map; the Levi-Civita connection in (pseudo)riemannian geometry, curvature quantities and identities; invariant metrics on Lie groups, model spaces; integration, Haar measure, semisimple groups and the Cartan-Killing form, biinvariant metrics on compact groups; submanifolds and relative curvature, the extrinsec Gauss-Bonnet Theorem for hypersurfaces.
Grading: Weakly competitive. I will assign problems in each lecture, ranging in difficulty from routine to rather challenging. Solving two thirds of them will be recommended and considered a perfect score.
Special Needs: If you have a physical, psychiatric, medical, or learning disability that could adversely affect your ability to carry out assigned course work, we urge you to contact the staff in the Disabled Student Services office (DSS), Room 133 Humanities, 632-6748/TDD. DSS will review your situation and determine, with you, what accomodations are necessary and appropriate. All information and documentation of disability is confidential.