TGTS Tetrahedral Geometry/Topology Seminar
Sponsored by Elizabethtown College, Franklin and Marshall College,
Lebanon Valley College, Millersville University, and West Chester University
Speakers Spring 2025
7 Feb 2025: Vince Coll, Lehigh University
7 Mar 2025: David Johnson, Lehigh University
4 Apr 2025: Subhajit Mishra, McMaster University
TGTS is a regional mathematics seminar/colloquium. We meet at 4:30
p.m. on the first Friday of each month during the academic year (with
some exceptions, as noted in the schedule above). The public is
cordially invited to attend.
Next Talk
Date/Time: 7 March 2025, 4:30pm ET
Location: Room 200, Wickersham Hall, Millersville University
Zoom: (zoom link to talk, Passcode: 804813)
Speaker: David Johnson, Lehigh University
Title: Orthogonal coordinates on (real) 4-dimensional Kähler manifolds
Abstract: C. F. Gauss constructed coordinates on any surface in
space so that $F=0$, that is, so that the coordinate directions were
orthogonal in a neighborhood. In 1984, Dennis DeTurck and Dean Yang
showed the existence of orthogonal coordinates on any Riemannian
3-manifold.
They also showed that, for dimensions at least 4, there is a curvature
obstruction to the existence of orthogonal coordinates, in that
curvature components of the form $R_{ijkl}$, with all 4 indices
distinct, will vanish if the directions correspond to orthogonal
coordinates.
Recently, Paul Gauduchon and Andrei Moroianu showed that there are no
orthogonal coordinates on $\mathbb{CP}^{n}$ or $\mathbb{HP}^{n}$, with
the standard metrics, if $n>1$. In the case of $\mathbb{CP}^{2}$, the
curvature condition is inadequate to show their result; a mysterious
trick is used instead.
Today's talk will focus on 4 (real)-dimensional Kähler manifolds, their
elegant and special curvature, and how the underlying complex-analytic
structure lies behind Gauduchon and Moroianu's result, which reveals
further obstructions to the existence of orthogonal coordinates.
Past Seminars
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