Riemannian manifolds with homogeneous geodesics.

*(English)*Zbl 0731.53046This paper contains a classification of g.o. spaces in dimension \(n\leq 6\). A Riemannian homogeneous space is called a g.o. space if each geodesic is an orbit of a one-parameter group of isometries. Naturally reductive spaces are g.o. spaces, whereas the first examples of g.o. spaces which are in no way naturally reductive were given by A. Kaplan, with the so-called generalized Heisenberg groups with two-dimensional centers.

The authors give a suitable definition of “infinitesimal isotropy type” for a g.o. space and a definition of natural reductivity for a Lie algebra \(\mathfrak h\subseteq \mathfrak{so}(k)\) acting on \(\mathbb R^k\) with \(\mathrm{Ker}\,h=\{0\}\). Furthermore, they state a list of naturally reductive algebras on \(\mathbb R^n\), in low dimension, and looking at the infinitesimal isotropy type of a g.o. space \((M,g)\), they obtain the following results: All Riemannian g.o. spaces of dimension \(n\leq 4\) are naturally reductive. Each 5-dimensional Riemannian g.o. space is naturally reductive or of isotropy type \(\mathrm{SU}(2)\). In the last case \((M,g)\) can be expressed as a homogeneous space of isotropy type \(\mathrm{U}(2)\) which is already naturally reductive. Finally, in dimension six, the authors give the explicit list of all simply-connected g.o. spaces which are not naturally reductive in any group extension. They find two classes of such spaces which are also 4-symmetric spaces. One class consists of compact spaces and the other one consists of two-step nilpotent Lie groups with two-dimensional centers and special left-invariant metrics.

The authors give a suitable definition of “infinitesimal isotropy type” for a g.o. space and a definition of natural reductivity for a Lie algebra \(\mathfrak h\subseteq \mathfrak{so}(k)\) acting on \(\mathbb R^k\) with \(\mathrm{Ker}\,h=\{0\}\). Furthermore, they state a list of naturally reductive algebras on \(\mathbb R^n\), in low dimension, and looking at the infinitesimal isotropy type of a g.o. space \((M,g)\), they obtain the following results: All Riemannian g.o. spaces of dimension \(n\leq 4\) are naturally reductive. Each 5-dimensional Riemannian g.o. space is naturally reductive or of isotropy type \(\mathrm{SU}(2)\). In the last case \((M,g)\) can be expressed as a homogeneous space of isotropy type \(\mathrm{U}(2)\) which is already naturally reductive. Finally, in dimension six, the authors give the explicit list of all simply-connected g.o. spaces which are not naturally reductive in any group extension. They find two classes of such spaces which are also 4-symmetric spaces. One class consists of compact spaces and the other one consists of two-step nilpotent Lie groups with two-dimensional centers and special left-invariant metrics.

Reviewer: Anna Maria Pastore (Bari)

##### MSC:

53C30 | Differential geometry of homogeneous manifolds |

53C22 | Geodesics in global differential geometry |