Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups

“…In this section, we show that an expression of the moduli space PM (p,q) derives a Milnor-type theorem for left-invariant pseudo-Riemannian metrics on G. The story is analogous to the arguments of [6], in which Milnor-type theorems for left-invariant Riemannian metrics have been studied.…”

“…Both of the Riemannian and pseudo-Riemannian cases are interesting, but sometimes the properties and methodologies are different. Recently many results on the Riemannian case have been obtained (just as examples, see [5,6,8,19] and references therein). However, the studies on the pseudo-Riemannian case seem to be still developing.…”

“…For every inner product , of signature (p, q) on g, there exists a basis of g, which is pseudo-orthonormal with respect to , up to scalar, and the bracket relations among the elements of the basis can be written with relatively smaller number of parameters. This kind of theorem is called a Milnor-type theorem in [6]. The name comes from the famous result by Milnor ([12]), who obtained this kind of theorems for left-invariant Riemannian metrics on all three-dimensional unimodular Lie groups…”

“…Milnor-type theorems have also been known for left-invariant Riemannian metrics on Lie groups with dim ≤ 4 ( [2,4,9,10]), and for left-invariant pseudo-Riemannian (Lorentzian) metrics on Lie groups with dim = 3 ([3, 17, 18]). Recently, for left-invariant Riemannian metrics, a general procedure to obtain Milnor-type theorems has been formulated in [6]. It is based on the moduli space of left-invariant Riemannian metrics on G, where the moduli space is defined as the orbit space of the action of R × Aut(g) on the space M(G) of left-invariant Riemannian metrics on G. In fact, an expression of the moduli space derives a Milnor-type theorem for G.…”

Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

“…Preliminaries on Milnor-type theorems. In this subsection, we recall a method for studying all inner products on a given Lie algebra g. This method is called a Milnor-type theorem in [6], since it generalizes the famous theorem by Milnor ([19]).…”