MAT 696
Mathematics Seminar

Stony Brook, Spring 2005
Last update: April 15

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

    Detlef Gromoll, Math 5-110
Phone: 632-8286,   Email:
Office Hours: Th 1:30-2:30 (5-110), and by appointment
    Wolfgang T. Meyer, Math 4-111
Phone: 632-8273,   Email:
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.

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)=RP3 and Spin(3)=S3.]

          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 H3; 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)