Most of my research has been on the theory and applications of
pseudoRiemannian geometries. (*Some
History of Geometry *tries to explain these in layman's terms
and put my work into a historical perspective.) In the earlier part of my
career, this was centered in applications via Lorentzian geometry to
General Relativity. More recently, the focus has shifted to two main
themes. One of these is understanding the crucial differences between the
more classical and much better understood Riemannian geometries, and the
more applicable but still poorly understood pseudoRiemannian geometries.
In conjunction with my collaborators, I have made several significant
contributions to this area. This was recognized when the editors of
Elsevier's *Encyclopedia of Mathematical Physics *asked me to
write the article on them.

Left-invariant metrics on Lie groups are the tightest combination of the geometries of Riemann and Klein. Traditionally, only some of these metric geometries were studied: the Riemannian (positive-definite) ones. Unfortunately, little is yet known about pseudoRiemannian left-invariant metrics on Lie groups, and only the Lorentzian case has been studied in any generality. In dimension 3, Cordero and I determined all possible left-invariant metric geometries on Lie groups. In this dimension there are only two main types: Riemannian and Lorentzian; but in each higher dimension, there are many types of indefinite metrics while there is essentially only one type of definite metric. The problem on Lie groups in particular is more complicated than merely sorting by signature: the degeneracy of the center also plays a crucial role. For example, the 3-dimensional Heisenberg group has two different classes of Lorentzian metrics, while Eberlein has shown that there is only one class of Riemannian metric (determined up to a scaling factor).

Some of the most tractable Lie groups are the nilpotent Lie groups: they are the closest to being commutative. The left-invariant metric geometries on commutative Lie groups are Euclidean geometry and its pseudoEuclidean relatives. Thus the left-invariant metric geometries on nilpotent Lie groups are those that are closest to being familiar while still exhibiting distinctive new features.

Among the nilpotent Lie groups, the very nearest to commutative are the
2-step ones. In recent years, some of the most exciting new results in
definite metric geometries have been obtained with 2-step nilpotent Lie
groups; *e.g., *the isospectral but nonisometric examples of C.
Gordon and her collaborators. The left-invariant Riemannian geometry of
these was studied in some detail by Eberlein, and is a focus of continuing
research by him, Mast, Gornet, and many others.

Cordero and I studied left-invariant pseudoRiemannian geometries on 2-step
nilpotent Lie groups. This was another step in my long-term program to
increase significantly our knowledge of the general features of
pseudoRiemannian geometries. Other recent parts of this program included
other joint work with Cordero, with Beem and Low, and with Jang.
Left-invariant pseudoRiemannian geometries on 2-step nilpotent Lie groups
exhibit several new phenomena that do not occur in the Riemannian case;
*e.g., *flat metrics always exist. Some of these provide links
to other areas of current mathematical research, such as splitting of
foliations or decoupling of systems of differential equations, thereby
enriching much more than just geometric study.

Building on that work, Jang and I gave the first complete determination of
the conjugate locus in two classes of pseudoRiemannian 2-step nilpotent
Lie groups. We found all conjugate points on geodesics parallel and
perpendicular to the center, and an explicit formula for the complete
conjugate locus when the center is 1-dimensional. It is noteworthy that
the latter included the first publication of the Riemannian case itself.
We also obtained new information on the analytic structure of the
conjugate locus in general. The next step is to be a determination of the
exact level of degeneracy, if any, of all these conjugate points according
to the recent classification by Piccione *et al. *

The other main theme is the generalization of the theory of systems of geodesics to general second-order differential equations over smooth manifolds. This began as joint work with Del Riego. The underlying geometric structure of geodesic systems (homogeneous quadratic sprays) is a linear connection. Going to higher-order systems of curves introduces minimal nonlinearities and allows the theory to be applied to systems of bicharacteristics of pseudodifferential operators of arbitrary order. We made good progress on the study of fully nonlinear systems. I extended the fundamental constructions of the exponential map (only now there is a 1-parameter family of exponential maps), associated Ehresman connections (possibly nonlinear), and covariant derivatives (possibly nonlinear, but still in bijective correspondence with Ehresmann connections), and of Jacobi fields (which now come in longitudinal as well as the usual transverse type). These (possibly nonlinear) connections are of interest in certain areas of physics (such as nonlinear quantum scattering). One of the major results is an extension of the Ambrose-Palais-Singer correspondence to Ehresmann connections and their associated geodesic systems.

At present, I am continuing the work on Jacobi fields with an eye toward extending the Ambrose-Singer correspondence. Work on parallelism and holonomy groups for Ehresmann connections is also beginning. Several more papers seem reasonable here.

A survey of the entire program of which all of these are parts may be found in my paper from the Beemfest.

- Arithmetic of Lorentzian 3-manifolds, especially the many relations
between their geometry and the arithmetic of
*Q(r)*, where*r*denotes the positive solution of*x² =*2. There are theta-functions on nonlinear lattices derived from sectional curvature on one hand, and an Apollonian circle-packing problem quite accessible to undergraduates on another. An introduction to the nonlinear lattices is in two papers with Fox [FP1, FP2], the first of which was published. - Sectional curvature as a rational morphism/function as in algebraic
geometry. This includes its functoriality and its three types of values:
regular, pole, and blowing-up. An early paper
with Beem shows some of these ideas in embryonic form. A bit more is
here.
- "Multipole" expansion of the sectional curvature function. Some
3-dimensional examples were presented in various talks and involve
Legendre functions of the first and second kinds and separable Hilbert
bundles over 3-manifolds. Here's something anent
the first area and this.
- There should be a Poisson summation formula (in fact, a complete L² theory) for the Fourier transform on the Poincaré disc for rapidly decreasing modular functions. (In case you wonder, the disc is a model for the hyperbolic plane and there is a relation with the sectional curvature of Lorentzian 3-manifolds.)