### Phil Parker, Professor

Differential Geometry, Math Physics; PhD, Oregon State University, 1977

### PhD Students

Paul D. Sinclair, "Metrics on Bundle Spaces and Harmonic Gauss Map", PhD thesis, 1991

### Research

There are two main lines, one around pseudoRiemannian (indefinite metric tensor) geometries and one around (possibly nonlinear) connections associated to second-order differential equations (quasisprays). The former has lately concentrated on 2-step nilpotent Lie groups (see here for a nontechnical, historical overview), most recently on the conjugate locus and degeneracies. The latter includes a major extension of the Ambrose-Palais-Singer correspondence; see here for a more technical overview with some history.

### Selected Publications

- L. Del Riego and P.E. Parker, Geometry
of Nonlinear Connections,
*Nonlinear Anal. * **63** (2005)
e501-e510.
- C. Jang, P.E. Parker, and K. Park, Pseudo
*H*-type 2-step Nilpotent Lie Groups, *Houston J. Math.* **31** (2005)
765-786.
- P.E. Parker, Geometry
of Bicharacteristics, in
*Advances in Differential Geometry and General
Relativity,* eds. S. Dostoglou and P. Ehrlich. Contemp. Math. 359.
Providence: AMS, 2004. pp.31-40.
- P.E. Parker, Pseudo-Riemannian Nilpotent Lie Groups, in
*Encyclopedia of Mathematical Physics.* eds. J.-P. Françoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol. 4, pp. 94-104.
- L.A. Cordero and P.E. Parker, Lattices and Periodic Geodesics in Pseudoriemannian 2-step Nilpotent Lie Groups,
*Int. J. Geom. Methods Mod. Phys.* **5** (2008) 79--99.
- L.A. Cordero and P.E. Parker, Isometry
Groups of pseudoRiemannian 2-step Nilpotent Lie Groups,
*Houston J.
Math.* **35** (2009) 49-72.