The Riemannian manifolds with boundary and large symmetry

TL;DR

This paper extends classical results on the symmetry groups of Riemannian manifolds to those with boundary, establishing an upper bound on the isometry group's dimension and classifying manifolds that achieve this maximum.

Contribution

It proves that the isometry group of a Riemannian manifold with boundary has a specific maximal dimension and provides a complete classification of manifolds attaining this bound.

Findings

Isometry group dimension is at most half of the manifold's dimension times (dimension minus one).

Complete classification of manifolds with boundary whose isometry group reaches the maximum dimension.

Extension of classical symmetry results to manifolds with boundary.

Abstract

Sixty years ago, S. B. Myers and N. E. Steenrod ({\it Ann. of Math.} {\bf 40} (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I. Zhukova ({\it Siberian Math. J.} {\bf 48} (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we firstly show that the isometry group of a Riemannian manifold $M$ with boundary has dimension at most ${1/2} \dim M (\dim M-1)$. Then we completely classify such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension.