Mathematics and Statistics Department
Phil Parker
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 and P.E. Parker, Conjugate Loci of PseudoRiemannian 2-step Nilpotent Lie Groups with Nondegenerate Center, Ann. Global Anal. Geom. 28 (2005) 1-18.
- C. Jang and P.E. Parker, Examples of Conjugate Loci of PseudoRiemannian 2-step Nilpotent Lie Groups with Nondegenerate Center, in Recent Advances in Riemannian and Lorentzian Geometries, eds. K.L. Duggal and R. Sharma. Contemp. Math. 337. Providence: AMS, 2004. pp.91-108.
- C. Jang, P.E. Parker, and K. Park, PseudoH-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.Fran�ise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol. 4, pp. 94-104.