Mathematics Seminar

Last update: April 15

*This page will be updated regularly. Check for announcements and postings!*

**Coordinators:**

Detlef Gromoll, Math 5-110

Phone: 632-8286,
Email: *detlef@math.sunysb.edu*

Office Hours: Th 1:30-2:30 (5-110), and by appointment

Wolfgang T. Meyer, Math 4-111

Phone: 632-8273,
Email: *wmeyer@math.sunysb.edu*

Office Hours: Th 1:30-2:30 (4-111), and by appointment

**About this Course:** The Graduate Mathematics Seminar will be offered jointly by Detlef Gromoll and Wolfgang Meyer (who is visiting this semester from the University of Muenster/Germany).

Progress in understanding the interaction of curvature and topology in global differential geometry has always hinged on examples in an exceptional way. Complete spaces with various basic curvature constraints are typically quite difficult to construct. We want to discuss some aspects of the broad problem concerning examples, mostly in the riemannian case. There have been exciting new developments since we gave a related course in the late nineties.

The topics are flexible; we can try to accomodate additional requests. However, from our experience, basic constructions will already take considerable time. We plan to touch on: Riemannian submersions, invariant metrics on Lie groups, homogeneous and symmetric spaces, double quotients: Eschenburg's examples, general multiple warping and glueing: examples of Cheeger, Shah-Yang,Wei etc., Sasakian metrics, cohomogeneity 1 manifolds: Grove-Ziller examples, Wilking's constructions, and more. Obviously some (but not necessarily an extensive) background in riemannian geometry is necessary; a few gaps can be filled as needed. Talks to be given by everyone, but auditing ok; S/U grading. Material will be taken from many diverse references; no particular text book.

**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.stonybrook.edu/facilities/ehs/fire/disabilities.shtml*

**Week by Week Details:** Seminar sessions will be listed here. There may or may not be details as to topics and references.

February 01 - Organizational meeting; general overview (Detlef Gromoll)

February 08 - *Geometry of left invariant metrics: Examples* I (Andrew Clarke)

[Elementary properties of left invariant and biinvariant metrics;

general methods for calculating curvatures and some simplifying formulas;

discussion of the most classical groups SO(3)=RP^{3} and Spin(3)=S^{3}.]

February 15 - *Geometry of left invariant metrics: Examples* II (Andrew Clarke)

[Further discussion of invariant metrics in the 3-dimensional case,

notably also on the Heisenberg group H^{3}; questions involving the

geometry of geodesics, time permitting.]

February 22 - *Riemannian submersions* I (Detlef Gromoll)

[A rapid course on the basic geometry of riemannian submersions:

Metric properties, curvature relations, transversal holonomy -

with first applications and examples concerning nonnegatively curved spaces.]

March 01 - *Riemannian submersions* II: *General warping* (Detlef Gromoll)

[We continue our introduction to the geometry of riemannian submersions and

metric foliations. After a brief discussion of curvature relations, we look

at the relatively accessible situation of metric fibrations of euclidean

space. We will then turn to simple and multiple warping techniques that

play an important role in constructions of examples of nonnegatively curved

spaces. We will describe basic applications and typical computations.]

March 15 - *Riemannian submersions* III:

*Applications of warping techniques* (Detlef Gromoll)

[This is the last of three lectures on the basic geometry of riemannian

submersions and metric foliations. We will summarize how curvature is

affected by warping a submersion. This helps, for example, understanding

collapsing spaces as well as constructing various classes of manifolds with

curvature constraints. We will focus on two specific applications - one where

controlling all sectional curvatures is the issue, the other concerning

positive ricci curvature.]

March 29 - *Normal and other homogeneous
spaces* (Stanislav Ostrovsky)

[A manifold is homogeneous if it admits a transitive action of a Lie

group G allowing to write it as quotient G/H. After a quick review,

we will discuss what it means for invariant metrics to be normal and

derive Nomizu's special curvature formula from the O'Neill Formula. We will

then look at examples, in particular the Berger Sphere, and briefly

describe the classification of normal homogeneous spaces that are strictly

positively curved. Finally, we will touch on the Wallach spaces in

dimension 7, important examples of homogeneous metrics that are not normal.]

April 12 - *Symmetric spaces; the* CROSS *case* (Andrew Bulawa)

[We will start with an introduction to symmetric spaces. Then we will

discuss an algebraic approach for curvature computations on such objects,

and explore geometric properties of some specific compact symmetric

spaces such as complex projective space.]

April 19 - *Biquotients; Eschenburg and Bazaikin examples* (Wolfgang Meyer)

[All examples of manifolds admitting metrics of positive curvature that

have been discussed so far were homogeneous spaces. In this lecture we

we discuss another class of examples that arise as bi-quotients of Lie

groups. In contrast to homogeneous spaces there is no hope for a

classification of bi-quotients as yet. On the other hand, there are

very interesting examples, among them an exotic 7-sphere with nonnegative

(almost positive) curvature, and infinite families of positively curved

bi-quotients in dimensions 7 and 13 that are not homotopically equivalent

to homogeneous spaces.]

April 26 - *Cohomogeneity* 1 *manifolds; Grove-Ziller examples* I (Matthew Kudzin)

May 03 - *Cohomogeneity* 1 *manifolds* II (tentative) (Matthew Kudzin)