The Lie group of isometries of a pseudo-Riemannian manifold

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.

Keywords

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

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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