Phil Parker, Professor
Differential Geometry, Math Physics; PhD, Oregon State University, 1977
Justin Ryan, "Geometry of Horizontal Bundles and Connections," 2014
Paul D. Sinclair, "Metrics on Bundle Spaces and Harmonic Gauss Maps", 1991
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.
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. 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.
L. Del Riego and P. E. Parker, General connections, Exponetial Maps, and Second order Differential Equations, Differ, Geom, Dyn Syst. 13 (2011) 72-90.