Patrick Eberlein's Home Page
I am a Professor in the Mathematics
Newton's Method in Two Variables PDF version
A.B. Harvard 1965
M.A. UCLA 1967
Ph.D. UCLA 1970 (Adviser : Barrett O'Neill)
Current information as of September, 2005
How to find me.
My office is in Phillips Hall 392. My office phone number is: (919) 962-9624.
My Fall 2005 office hours are:
In Ph 392 :
Monday 1 - 2 PM
Tuesday 4:15-5:15 PM
Wednesday 4:15-5:15 PM
Thursday 1 - 2 PM
or by appointment
You can also email me at firstname.lastname@example.org.
My Fall 2005 teaching schedule
Math 33A, MWF 12-1 PM, Phillips 367
Math 33A Fall 2005 PDF version
Math 33A Fall 2005 PDF version
My research interests
I work in Riemannian geometry. Most of my work concerns
the structure of manifolds with nonpositive sectional curvature. The simply
connected model spaces are the symmetric spaces, which have the property
that for every point of the space there exists an isometry that fixes the
point and reverses all geodesics through the point (i.e. there exists a
reflection through each point of the space). Symmetric spaces can be represented
as coset spaces G/K, where G is a connected semisimple group with no compact
factors and K is a maximal compact subgroup of G.
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
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.
I am a member of the AMS and MAA.
Averaging over compact groups PDF version
Clifford algebras I PDF version
Clifford algebras II PDF version
Clifford algebras III PDF version
Clifford algebras IV PDF version
Semisimple rings and modules PDF
Rigid motions of Euclidean space PDF
Lie triple systems PDF version
The manifold structure of G/H PDF
The Ricci tensor and Killing vector fields in Riemannian symmetric spaces
Click here for a Complete
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
The moduli space of 2-step nilpotent Lie algebras of type (p,q), Contemp.
Math. 332 (2003), 37-72
Left invariant geometry of Lie groups, Cubo 6(2004), no.1,427-510.
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
Rational approximation in compact Lie groups and their Lie algebras, preprint,2000
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),
Geodesic flows on negatively curved manifolds,I, Annals of Math. 95(1972),
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.
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