Date/Time: 7 March 2025, 4:30pm ET
 Location: Room 200, Wickersham Hall, Millersville University
     Zoom: (zoom link to talk)
  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.