For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense that the bracket relations among them can be written with relatively smaller number of parameters. Our procedure is based on the moduli space of left-invariant Riemannian metrics. Some explicit examples of such frames and applications will also be given.
Abstract. In this paper, we define the corresponding submanifolds to leftinvariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.
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