Special ball-homogeneous spaces. (English) Zbl 0892.53023

Calvaruso, G.; Vanhecke, Lieven

Z. Anal. Anwend. 16, No. 4, 789-800 (1997).

A ball-homogeneous space is a Riemannian manifold \((M,g)\) on which the volumes of all sufficiently small geodesic balls only depend on the radius. In dimensions greater than two, it is an open problem whether each ball-homogeneous Riemannian manifold is locally homogeneous. In the present paper a positive answer is given in the following special situations: (a) \(M\) is three-dimensional with at most two distinct Ricci eigenvalues; (b) \(M\) is three-dimensional and its Ricci tensor is either a Codazzi tensor, or it is cyclic parallel; (c) \(M\) is conformally flat with at most three distinct Ricci eigenvalues (and of arbitrary dimension).

Reviewer: Oldrich Kowalski (Praha)

MSC:

53C30	Differential geometry of homogeneous manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C35	Differential geometry of symmetric spaces

Keywords:

conformal flatness; ball-homogeneous space; Ricci tensor; distinct Ricci eigenvalues

References:

[1]	Abbena, E., Garbiero, S. and L. Vanhecke: Einstein-like metrics on three-dimensional Riemannian homogeneous manifolds. Simon Stevin 66 (1992), 173 - 182. · Zbl 0786.53028
[2]	Blair, D. E.: Contact Manifolds in Riemannian Geometry. Lect. Notes Math. 509 (1976), 1 -. 146. · Zbl 0319.53026
[3]	Boeckx, E., Kowalski, 0. and L. Vanhecke: Riemannian Manifolds of Conullity Two. Singapore: World Sci. Publ. 1996. · Zbl 0904.53006
[4]	Calvaruso, C., Tondeur, Ph. and L. Vanhecke: Four-dimensional ball-homogeneous and C-spaces. Beitr. AIg. Geom. (to appear). · Zbl 0884.53037
[5]	Calvaruso, G. and L. Vanhecke: Semi-symmetric ball-homogeneous spaces and a volume conjecture. Bull. Austral. Math. Soc. (to appear). · Zbl 0903.53031
[6]	Deprez, J., Deszcz, R. and L. Verstraelen: Examples of pseudo-symmetric conformally flat warped products. Chinese J. Math. 17 (1989), 51 - 65. · Zbl 0678.53022
[7]	Garcia-Rio, E. and L. Vanhecke: Five-dimensional co-symmetric spaces. Balkan J. Geom. App!. (to appear).
[8]	Gray, A.: Geodesic balls in Riemannian product manifolds. In: Differential Geometry and Relativity (Math. Phys. and App!. Math: Vol. 3; eds.: M. Cahen and M. Flato). Dordrecht: Reide! Pub!. Co. 1976, pp. 63 - 66. · Zbl 0345.53007
[9]	Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicata 7 (1978), 259 - 280. · Zbl 0378.53018
[10]	Gray, A. and L. Vanhecke: Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142 (1979), 157 - 198. · Zbl 0428.53017
[11]	Gunther, P. and F. Prüfer: D’ Atri spaces are ball-homogeneous. Preprint 1995.
[12]	Kowalski, 0.: A classification of Riemannian 3-manifolds with constant principal Ricci curvatures g = g3. Nagoya Math. J. 132 (1993), 1 - 36. · Zbl 0788.53038
[13]	Kowalski, 0.: An explicit classification of 3-dimensional Riemannian spaces satisfying Rxy . R = 0. Czechoslovak Math. J. 46(121) (1996), 427 - 474. · Zbl 0879.53014
[14]	Kowalski, 0., Prüfer, F. and L. Vanhecke: D’ Atri spaces. In: Topics in Geometry: In Memory of Joseph D’ Atri (Progr. Nonlin. Duff. Equ.: Vol. 20; ed.: S. Gindikin). Boston - Base! - Berlin: Birkhäuser Verlag 1996, pp. 241 - 284. · Zbl 0862.53039
[15]	Kowalski, 0. and L. Vanhecke: Ball-homogeneous and disk-homogeneous Riemannian manifolds. Math. Z. 180 (1982), 429 - 444. · Zbl 0476.53023
[16]	Perrone, D.: Varietà conformemente piatte e geometria spettrale. Riv. Mat. Univ. Parma 4 (1982), 317 - 330. · Zbl 0514.53041
[17]	Perrone, D.: Homogeneous contact Riemannian three-manifolds. Preprint 1997. · Zbl 0906.53031
[18]	Prüfer, F., Tricerri, F. and L. Vanhecke: Curvature invariants, differential operators and local homogeneity. Trans. Amer. Math. Soc. 348 (1996), 4643 - 4652. · Zbl 0867.53032
[19]	Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isome- tries. Tôhoku Math. J. 27 (1975), 103 - 110. · Zbl 0311.53062
[20]	Vanhecke, L.: Scalar curvature invariants and local homogeneity. In: Proc. Workshop on Differential Geometry and Topology (dedicated to the memory of Franco Tricerri), Palrmo 1996 (to appear). · Zbl 0894.53046
[21]	Yainato, K.: A characterization of locally homogeneous Riemann manifolds of dimension 3. Nagoya Math. J. 123 (1991), 77 - 90. · Zbl 0738.53032