Journal of the Mathematical Society of Japan

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

Takahiro HASHINAGA, Hiroshi TAMARU, and Kazuhiro TERADA

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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.

Article information

J. Math. Soc. Japan, Volume 68, Number 2 (2016), 669-684.

First available in Project Euclid: 15 April 2016

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Zentralblatt MATH identifier

Primary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

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


HASHINAGA, Takahiro; TAMARU, Hiroshi; TERADA, Kazuhiro. Milnor-type theorems for left-invariant Riemannian metrics on Lie groups. J. Math. Soc. Japan 68 (2016), no. 2, 669--684. doi:10.2969/jmsj/06820669.

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