Milnor-type theorems for left-invariant Riemannian metrics on Lie groups. (English) Zbl 1353.53058

Hashinaga, Takahiro; Tamaru, Hiroshi; Terada, Kazuhiro

J. Math. Soc. Japan 68, No. 2, 669-684 (2016).

The authors prove the existence of special left-invariant orthonormal frames for left-invariant Riemannian metrics on Lie groups. This means that the bracket relations among them can be written with a relatively smaller number of parameters. This result could be considered as a generalization of well-known results of J. W. Milnor on 3-dimensional unimodular Lie groups [Adv. Math. 21, 293–329 (1976; Zbl 0341.53030)]. The main tool is based on the structure of the moduli space of left-invariant Riemannian metrics. The authors consider some explicit examples of such frames and some applications.

Reviewer: Yurii G. Nikonorov (Volgodonsk)

MSC:

53C30	Differential geometry of homogeneous manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Lie groups; left-invariant Riemannian metrics; Milnor frames; Milnor-type theorems; Ricci signatures; solvsolitons

Citations:

Zbl 0341.53030

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