Clifford-Wolf homogeneous Riemannian manifolds. (English) Zbl 1179.53043

Berestovskii, Valerii N.; Nikonorov, Yurii G.

J. Differ. Geom. 82, No. 3, 467-500 (2009).

An isometry \(f\) of a metric space \((X,d)\) is called a Clifford-Wolf translation if there exists a constant \(c\) such that \(d(x,f(x)) = c\) for all \(x \in X\). A metric space \((X,d)\) is said to be Clifford-Wolf homogeneous if for all \(x,y \in X\) there exists a Clifford-Wolf translation \(f\) on \(X\) with \(f(x) = y\). Elementary examples of Clifford-Wolf homogeneous metric spaces are Euclidean spaces, odd-dimensional round spheres, and Lie groups with biinvariant Riemannian metrics. Moreover, every direct metric product of Clifford-Wolf homogeneous metric spaces is again a Clifford-Wolf homogeneous metric space.
The main result of the authors states that for simply connected Riemannian manifolds the converse is true: A connected, simply connected Riemannian manifold is Clifford-Wolf homogeneous if and only if it is a direct metric product of a Euclidean space, an odd-dimensional sphere of constant sectional curvature, and simply connected Lie groups equipped with biinvariant Riemannian metrics (where some of these factors could be absent).
For the proof the authors develop techniques based on Killing vector fields of constant length. This allows them also to classify all simply connected Riemannian manifolds \((M,g)\) having the Killing property: For each point \(p \in M\) there exists a neighborhood \(U\) and an orthonormal frame field \(X_1,\dots,X_n\) of \(M\) along \(U\) consisting of Killing vector fields \(X_i\). The authors prove that a complete, simply connected Riemannian manifold has the Killing property if and only if it is a direct metric product of a Euclidean space, \(7\)-dimensional spheres of constant sectional curvature, and compact simply connected simple Lie groups equipped with biinvariant Riemannian metrics (where some of these factors could be absent).
The authors also study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss several connections between these spaces and some classical objects.

Reviewer: Jürgen Berndt (London)

MSC:

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

Keywords:

homogeneous Riemannian manifolds; Clifford-Wolf translations; Killing vector fields of constant length; Riemannian manifolds with Killing property; Clifford-Killing spaces