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A class of nonsymmetric harmonic Riemannian spaces
Author(s):
Ewa
Damek;
Fulvio
Ricci
Journal:
Bull. Amer. Math. Soc.
27
(1992),
139-142.
MSC (2000):
Primary 53C35
MathSciNet review:
1142682
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Abstract:
Certain solvable extensions of H-type groups provide noncompact counterexamples to a conjecture of Lichnerowicz, which asserted that "harmonic" Riemannian spaces must be rank 1 symmetric spaces.
References:
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- [1]
- A. L. Besse, Manifolds all of whose geodesics are closed, Springer, Berlin, 1978. MR 496885 (80c:53044)
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- J. Boggino, Generalized Heisenberg groups and solvmanifolds naturally associated, Rend. Sem. Mat. Univ. Polit. Torino 43 (1985), 529-547. MR 884876 (88f:53093)
- [3]
- M. Cowling, A. H. Dooley, A. Korányi, and F. Ricci, H-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), 1-41. MR 1102963 (92e:22017)
- [4]
- -, H-type groups and Iwasawa decompositions, II (to appear).
- [5]
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- [6]
- -, Curvature of a semidirect extension of a Heisenberg type nilpotent group, Colloq. Math. 53 (1987), 249-253. MR 924069 (89d:22007)
- [7]
- E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. (to appear). MR 1164603 (93d:43006)
- [8]
- A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by compositions of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. MR 554324 (81c:58059)
- [9]
- A. Korányi, Geometric properties of Heisenberg-type groups, Adv. Math. 56 (1985), 28-38. MR 782541 (86h:53050)
- [10]
- A. Lichnerowicz, Sur les espaces Riemanniens complètement harmoniques, Bull. Soc. Math. France 72 (1944), 146-168. MR 0012886 (7:80c)
- [11]
- Z. Szabó, The Lichnerowicz conjecture on harmonic manifolds, J. Differential Geom. 31 (1990), 1-28. MR 1030663 (91g:53052)
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Additional Information:
DOI:
10.1090/S0273-0979-1992-00293-8
PII:
S 0273-0979(1992)00293-8
Copyright of article:
Copyright
1992,
American Mathematical Society
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