Amalgamations of homogeneous Riemannian spaces. (English) Zbl 0563.53041

Tvarožek, Jozef

Čas. Pěst. Mat. 109, 355-371 (1984).

Amalgamation is a generalized product operation which enables to construct, for certain pairs of homogeneous Riemannian spaces \((M_ 1,g_ 1)\), \((M_ 2,g_ 2)\), a new homogeneous Riemannian space (M,g) such that 1) max (dim \(M_ 1,\dim M_ 2)<\dim M<\dim M_ 1+\dim M_ 2\); 2) for each \(i=1,2\), there is a totally geodesic foliation on (M,g) whose leaves are locally isometric to \((M_ i,g_ i)\). For instance, any two non-semi-simple Lie groups with arbitrary invariant metrics possess at least one amalgamation; and a lot of other examples can be given. [The basic idea comes from the reviewer, cf. Generalized symmetric spaces (1980; Zbl 0431.53042)]. Main theorem: If \((M_ 1,g_ 1)\), \((M_ 2,g_ 2)\) are two connected, simply connected and irreducible homogeneous Riemannian spaces, and if there exists an amalgamation (M,g), then (M,g) is always irreducible.

Reviewer: O.Kowalski

MSC:

53C30	Differential geometry of homogeneous manifolds
53C20	Global Riemannian geometry, including pinching

Keywords:

Amalgamation; totally geodesic foliation; irreducible homogeneous Riemannian spaces

Citations:

Zbl 0431.53042