zbMATH — the first resource for mathematics

D’Atri spaces. (English) Zbl 0862.53039
Gindikin, Simon (ed.), Topics in geometry. In memory of Joseph D’Atri. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 241-284 (1996).
The present paper is devoted to a survey of D’Atri spaces which were introduced by J. E. D’Atri and H. K. Nickerson [J. Differ. Geom. 9, 251-262 (1974; Zbl 0285.53019)] in their study of Riemannian and pseudo-Riemannian spaces whose local geodesic symmetries are volume-preserving, or equivalently divergence-preserving. Such spaces were subsequently shown to encompass a broad class of spaces which include special types of homogeneous, harmonic, and symmetric spaces. The survey is discursive and contains no proofs, but indicates open problems and directions for further research. Contents include: an introduction; preliminaries; characterizations of D’Atri spaces; special classes of D’Atri spaces; relations between the special classes; D’Atri spaces in low dimensions; D’Atri spaces, homogeneity, and nonpositive curvature; some modifications of the D’Atri property; D’Atri spaces in special geometries; volume of tubes about curves in D’Atri spaces; generalizations; and finally a 136 item bibliography. The paper is a valuable contribution to the literature and will be required reading for anyone interested in the subject.
For the entire collection see [Zbl 0842.00040].

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