×

zbMATH — the first resource for mathematics

A classification of special Riemannian 3-manifolds with distinct constant Ricci eigenvalues. (English) Zbl 0821.53036
This paper is a contribution to the explicit description of the metrics of a three-dimensional curvature homogeneous Riemannian space, that is, a Riemannian manifold with constant Ricci eigenvalues \(\rho_ i\), \(i = 1,2,3\). The case of equal eigenvalues is well known and the case \(\rho_ 1 = \rho_ 2 \neq \rho_ 3\) has been treated in detail by the first author. Here, the authors treat the case of distinct eigenvalues and obtain an explicit description under two additional geometrical assumptions by solving explicitly an associated system of first-order partial differential equations.

MSC:
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [3) Kobayashi, S. and K. Nomizu: Foundations of Differential Geometry. Vol. I. New York: Intersci. PubI. 1963. (4) Kowalski, 0.: An explicit classification of 3-dimensional Riemannian spaces satisfying R(X, Y) R = 0. Preprint 1991.
[2] Kowalski, 0.: Nonhomogeneous Riemannian 3-manifolds with distinct constant Ricci eigenvalues. Comment. Math. Univ. Carolinae 34 (1993), 451 - 457. · Zbl 0789.53024 · eudml:118411
[3] Kowalski, 0. and F. Prüfer: On Riemannian 3-manifolds with iiitinct constant Ricci eigenvalues. Math. Ann. (1994), 17 - 28. · Zbl 0813.53020 · doi:10.1007/BF01450473 · eudml:165239
[4] Kowalski, 0., Tricerri F. and L. Vanhecke: New examples of non-homogeneous Rieman- nian manifolds whose curvature tensor is that of a Riemannian symmetric space. C. R. Acad. Sci. Paris (Sér. 1)311(1990), 355- 360. · Zbl 0713.53028
[5] Kowalski, 0., Tricerri, F. and L. Vanhecke: Curvature homogeneous Riemannian mani- folds. J. Math. Pures AppI. 71(1992), 471 -501. · Zbl 0836.53029
[6] Kowalski, 0., Tricerri F. and L. Vanhecke: Curvature homogeneous spaces with a solvable - Lie group as a homogeneous model. J. Math. Soc. Japan 44 (1992), 461 - 484. · Zbl 0762.53031 · doi:10.2969/jmsj/04430461
[7] Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(1976), 293 - 329. · Zbl 0341.53030 · doi:10.1016/S0001-8708(76)80002-3
[8] [13] Sekigawa, K.: On . some 3-dimensional curvature homogeneous spaces. Tensor (N.S.) 31 (1977), 87 - 97. · Zbl 0356.53016
[9] Singer, I. M.: Infinitesimally homogeneous spaces. Comm. Pure Appi. Math. 13 (1960), 685 - 697. (15] Spiro, A. and F. Tricerri: 3-dimensional Riemannian metrics with prescribed Ricci prin- cipal curvatures. J. Math. Pures Appi. (to appear). · Zbl 0171.42503
[10] [17] Yamato, K.: A characterization of locally homogeneous Riemannian manifolds of dimen- sion three. Nagoya Math. J. 123 (1991), 77 - 90. · Zbl 0738.53032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.