On some differential geometric characterizations of the center of a Lie group (Q1192427)

Two interesting geometric characterizations of the center of a Lie group are given in the following theorems. Theorem 1. Let \(G\) be a Lie group and \(H\) its connected subgroup. The following conditions are equivalent (1) \(H\) is contained in the center of \(G\); (2) \(H\) is totally geodesic with respect to any left invariant Riemannian metric on \(G\). Theorem 2. Let \(G\) be a nilpotent or compact Lie group. Then for each \(X\in g\) the following conditions are equivalent (1) \(X\) belongs to the center of \(G\); (2) the inequality \(\text{Ric}(X)\geq 0\) holds for any left invariant Riemannian metric on \(G\), where \(\text{Ric}(X)\) denotes the Ricci curvature in the direction \(X\).