# zbMATH — the first resource for mathematics

Metric fibrations of Lobachevsky-Bolyai space. (English) Zbl 0793.53031
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 223-230 (1992).
Let $$x$$, $$y$$, $$u$$, $$v$$ be coordinates for $$H^ 2\times H^ 2$$, where $$H^ 2$$ is the hyperbolic plane, $$x,y$$ $$(u,v)$$ are horocyclic coordinates in the first (second) factor. The authors introduce metric fibrations (i.e. partitions of a metric space by isometric and locally equidistant subsets) $H^ 2\times H^ 2= \bigcup_ c \text{Sol}_ c, \qquad \text{Sol}_ c= \bigcup_{a+b=c} M_{a,b}$ (where $$\text{Sol}_ c= (y+v=c\}$$ is an Sol-manifold from the 8 homogeneous 3- dimensional manifolds in Thurston classification, $$M_{a,b}= \{y=a, v=b\}$$ is called homogeneous horotorus) and compare them with the metric fibrations of the second author [Sov. Math., Dokl. 38, 202-205 (1989); translation from Dokl. Akad. Nauk SSSR 301, No. 6, 1301-1304 (1988; Zbl 0681.53006)] $H^ 2\times H^ 2= \bigcup_ c H_ c^ 3, \qquad H_ c^ 3= \bigcup_{a-b=c} M_{a,b}$ (in hyperbolic 3-space $$H_ c^ 3= \{y-v=c\}$$, $$M_{a,b}$$ is an horosphere). The paper also contains a survey on metric fibrations of $$R^ n$$ and $$H^ n$$.
For the entire collection see [Zbl 0764.00002].

##### MSC:
 53C12 Foliations (differential geometric aspects)