[Binghamton Geometry/Topology Seminar] Topology/Geometry Seminar this Thursday

[Tam Nguyen Phan]

  BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR


   Date: Thursday, February 12, 2015

   Time:  2.50 pm

   Place: Whitney Hall, Room 100E,  followed by coffee/tea in the
      Hilton Reading Room. Note the new location of the math department.


   Speaker: James Dibble (Rutgers University)
   Title: Totally geodesic maps into manifolds with no focal points

Abstract:
A classical result of Eells-Sampson is that every homotopy class of maps
between compact Riemannian manifolds, where the target has non-positive
sectional curvature, contains an energy-minimizing harmonic representative.
They proved this by inventing the harmonic map heat flow, the first
geometric flow defined on manifolds. Their work was refined by Hartman, who
proved the monotonicity of certain distance functions under the flow and
used this to deduce that that the space of harmonic maps in each homotopy
class is path-connected and that energy is constant on it. Applying an
identity that dates to the work of Bochner, Eells-Sampson also proved that,
when the domain has non-negative Ricci curvature, all harmonic maps are
totally geodesic.

It will be shown that, for domains with non-negative Ricci curvature, the
results of Eells-Sampson, along with certain qualitative consequences of
Hartman's results, generalize to energy-minimizing maps into manifolds with
no focal points. These are manifolds whose universal covers satisfy a
simple synthetic condition: For each point and each maximal geodesic, there
is a unique geodesic connecting them that intersects the latter
perpendicularly. By contrast with previous approaches, the proof uses
neither a geometric flow nor the Bochner identity for harmonic maps.


NOTE:  The seminar has a webpage where the semester's program is listed:

http://www2.math.binghamton.edu/p/seminars/topsem

It can also be linked from the Department's Home Page.
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