The Riemannian manifolds with boundary and large symmetry

Abstract

Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most 1 2 dim M (dim M − 1). Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.

Isometry groups of Alexandrov spaces

“…In section 7 we bound the dimension of the isometry group of an Alexandrov space with boundary and classify these spaces up to homeomorphism when the isometry group has maximal dimension. This extends to the Alexandrov setting the work carried out in [10] for Riemannian manifolds with boundary. In our case, there appears a non-manifold with boundary and isometry group of maximal dimension, namely, the cone over a real projective space.…”

“…We now extend to Alexandrov spaces results by Chen, Shi and Xu [10] for Riemannian manifolds with boundary. We follow their proof closely, noting that one must make appropriate modifications for it to work in the Alexandrov case.…”

“…( Proof of Theorem 7.3. By looking at level sets of the distance function to a component of ∂X, we observe, as in [10], that the action of Isom(X) on X is of cohomogeneity one. Thus the orbit space X * = X/ Isom(X) is homeomorphic to either [0, 1] or [0, 1).…”

“…The isotropy at the other endpoint must contain SO(n − 1) and thus can only be O(n − 1) or SO(n), yielding the desired spaces. The only case not appearing in [10] is the one in which the points 0 and 1 in X * ≃ [0, 1] have, respectively, isotropy O(n−1) and SO(n). This corresponds to the cone over RP n−1 .…”

“…Since the action of Isom(X) on X is of cohomogeneity one, the metric of X is quite restricted. This idea was examined for smooth manifolds in [10], where the authors proved that the metrics were warped over an interval. It is natural to expect that similar versions can be given for Alexandrov spaces, although with certain modification on the warping function to reflect the possible lack of smoothness.…”