M.A. UCLA 1967

Ph.D. UCLA 1970 (Adviser : Barrett O'Neill)

- My office is in Phillips Hall 392. My office phone number is: (919) 962-9624.
- My Fall 2005 office hours are:
- In Ph 392 :
- You can also email me at pbe@email.unc.edu.

Monday 1 - 2 PM

Tuesday 4:15-5:15 PM

Wednesday 4:15-5:15 PM

Thursday 1 - 2 PM

or by appointment

- Math 33A, MWF 12-1 PM, Phillips 367

Since 1990 I have studied the geometry of 2-step nilpotent, simply connected Lie groups with a Riemannian metric that is preserved by left translations. This area is closely related to spaces of nonpositive sectional curvature. An interesting class of examples arises from a representation of a compact, semisimple p-dimensional Lie algebra g on a real vector space U of dimension q. The direct sum n = U + g admits a canonical 2-step nilpotent Lie algebra structure whose commutator ideal is g. The Lie algebra n has a natural inner product < , > that is unique up to scaling on irreducible g-submodules of U. Let N be the simply connected Lie group with Lie algebra n and left invariant metric arising from < , >. Then the geometry of N and its quotient manifolds is described by the roots and weights of the complexifications of g and U.

- Two step nilpotent Lie groups arising from semisimple modules, preprint, January 2005 PDF version
- Stabilizer Lie Algebras, preprint, July 2003 PDF version
- Geometry of 2-step nilpotent Lie groups, Modern Dynamical Systems, Cambridge University Press, 2004,67-101 PDF version
- The moduli space of 2-step nilpotent Lie algebras of type (p,q), Contemp. Math. 332 (2003), 37-72 PDF version
- Left invariant geometry of Lie groups, Cubo 6(2004), no.1,427-510. PDF version
- Riemannian submersions and lattices in 2-step nilpotent Lie groups,Comm. Analysis and Geom., 11 (2003), 441-488 PDF version. See also Appendix 1 and Appendix 2
- Rational approximation in compact Lie groups and their Lie algebras, preprint,2000 PDF version
- Rational approximation in compact Lie groups and their Lie algebras, II, preprint, 2000 PDF version
- Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press,1996, 449 pages.
- (joint with J. Heber) Quarter pinched homogeneous spaces of negative curvature, Inter. Jour. Math.,7 (1996), 441-500.
- (joint with C. Croke and B. Kleiner) Conjugacy and rigidity for nonpositively curved manifolds of higher rank, Topology, 35 (1996),273-286.
- Geometry of 2-step nilpotent groups with a left invariant metric, Annales de l'Ecole Norm. Sup., 27 (1994),611-660.
- (joint with J. Heber) A differential geometric characterization of symmetric spaces of higher rank, Publ. IHES. 71 (1990), 33-44.
- Symmetry diffeomorphism group of a manifold of nonpositive curvature, II,Indiana Univ. Math. Jour. 37(1988), 735-752.
- (joint with W. Ballmann)The fundamental group of compact manifolds with nonpositive curvature, Jour. Diff. Geom. 25(1987), 1-22.
- (joint with M. Brin and W. Ballmann)Structure of manifolds of nonpositive curvature, I, Annals of Math. 122(1985), 171-203.
- Isometry groups of simply connected manifolds of nonpositive curvature, II, Acta Math. 149(1982), 41-69
- Lattices in manifolds of nonpositive curvature,Annals of Math. 111(1980),435-476.
- When is a geodesic flow of Anosov type ?, I, Jour. Diff. Geom. 8(1973), 437-463.
- Geodesic flows on negatively curved manifolds,I, Annals of Math. 95(1972), 492-510.
- Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167(1972), 151-170.
- Manifolds admitting no metric of constant negative curvature, Jour. Diff. Geom. 5(1971), 59-60.

For comments/suggestions about this page, mail pbe@email.unc.edu