Annals of Mathematical Sciences and Applications

Volume 8 (2023)

Number 2

Special issue dedicated to Anthony To-Ming Lau on his 80th birthday

Guest Editors: Xiaolong Qin, Ngai-Ching Wong and Jen-Chih Yao

The Lie group of isometries of a pseudo-Riemannian manifold

Pages: 223 – 238



Hung Pham (School of Mathematics and Statistics, Victoria University of Wellington, New Zealand)


We give an elementary proof of the Myers–Steenrod theorem, stating that the group of isometries of a connected Riemannian manifold $M$ is a Lie group acting smoothly on $M$. Our proof follows the approach of Chu and Kobayashi, but replacing their use of a theorem of Palais with a topological condition detecting when a locally compact subspace of $M$ is an embedded integral manifold of a given $k$-plane distribution.


Lie group, locally compact group, isometries, pseudo-Riemannian manifold, integral manifold

Dedicated to Professor Anthony To-Ming Lau with admiration on the occasion of his 80th birthday

Received 9 April 2023

Accepted 26 April 2023

Published 26 July 2023