Differential Geometry

Last update: April 20

*This page will be updated at least once a week. Check for announcements and postings regularly, mostly at the bottom of the page!*

**Lecturer:** Detlef Gromoll, Math 5-116

Phone: 632-8290,
Email: *detlef@math.sunysb.edu*

Classes: MW 10:15-11:35am, Mathematics P-131

Office Hours: M 3-4 (UG Office P-143), Th 1:30-2:30 (5-116), and by appointment

**Grader:**
Martin Reiris, Math 2-112

Email: *martinr@math.sunysb.edu*

**About this Course:** We will cover basic topics of classical and current (mostly) riemannian and metric geometry at an intermediate thorough level. We will also build up some Lie Theory and look at homogeneous and symmetric spaces on the way. The exact choice and emphasis of topics is somewhat flexible and can be taylored to the audience.

**Prerequisites:** Last semester's MAT 568 is not really a prerequisite. Some familiarity with the foundations will be helpful (see beginning of standard topics below), but we will fill in gaps as needed.

**Text:** No particular text book will be used - we rather provide diverse references whenever needed. Taking detailed notes will be useful.

**Grading/Homework:** Problem solving is essential. I will assign problems weekly on this page ranging in difficulty from routine to rather challenging. Turning in solutions to at least half of them will be considered a perfect score if your selection is varied. There will be no other exams.

**Standard Topics:** 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, geometry of left-invariant metrics on SO(3), differential forms and integration, Haar measure, semisimple Lie groups and the Cartan-Killing form, biinvariant metrics on compact groups; submanifolds and relative curvature.

**Advanced Topics:** Total curvature theorems; riemannian submersions, normal homogeneous spaces; first and second variation of arc length and energy, Jacobi fields, (locally) symmetric spaces; completeness; elements of comparison theory, curvature and global shape: How curvature restricts the metric and topological structure in the large; the quest for more examples, notably of nonnegatively curved spaces - cohomogeneity 2 manifolds (recent work of Grove, Ziller, Wilking).

**Special Needs:**
If you have a physical, psychological, medical or learning disability that
may impact your course work, please contact Disability Support Services,
ECC (Educational Communications Center) Building, room 128, (631) 632-6748.
They will determine with you what accommodations are necessary and
appropriate. All information and documentation is confidential.

Students requiring emergency evacuation are encouraged to discuss their
needs with their professors and Disability Support Services. For procedures
and information, go to the following web site.

*http://www.ehs.stonybrook.edu/fire/disabilities.asp*