Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 068, 6 pages      arXiv:2005.00161      https://doi.org/10.3842/SIGMA.2020.068
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces

Yukai Sun a and Xianzhe Dai b
a)  School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, P.R. of China
b)  Department of Mathematics, UCSB, Santa Barbara CA 93106, USA

Received May 04, 2020, in final form July 22, 2020; Published online July 25, 2020

Abstract
Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group $G$ are extremal (in fact rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact the same result holds for a compact connected homogeneous manifold $G/H$ with $G$ compact connect and semi-simple.

Key words: extremal/rigid metrics; Lie groups; homogeneous spaces; scalar curvature.

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