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Patrick Eberlein's Home Page


I am a Professor in the Mathematics Department.


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 Fall 2005 teaching schedule

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

Homework Assignments

Math 33A Fall 2005 PDF version

Homework Solutions

Math 33A Fall 2005 PDF version

Supplementary Material

  • Newton's Method in Two Variables 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 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.

    I am a member of the AMS and MAA.

    Course notes

  • 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 version
  • Rigid motions of Euclidean space PDF version
  • Lie triple systems PDF version
  • The manifold structure of G/H PDF version
  • The Ricci tensor and Killing vector fields in Riemannian symmetric spaces PDF version

  • Selected Publications

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



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