The isometry groups of Riemannian orbifolds

Abstract

We prove that the isometry group I(N ) of an arbitrary Riemannian orbifold N , endowed with the compact-open topology, is a Lie group acting smoothly and properly on N . Moreover, I(N ) admits a unique smooth structure that makes it into a Lie group. We show in particular that the isometry group of each compact Riemannian orbifold with a negative definite Ricci tensor is finite, thus generalizing the well-known Bochner's theorem for Riemannian manifolds.

The Riemannian manifolds with boundary and large symmetry

“…As long as the isometry group of a Riemannian orbifold is concerned, quite recently, Bagaev and Zhukova [1] showed the same result as Facts 1.1-1.2. They generalized the idea of Kobayashi to their setting by using the orthonormal frame bundle of a Riemannian orbifold.…”

“…In this paper, we consider a special class of orbifolds -manifolds with boundary. We firstly observe that the dimension of the isometry group I(M ) of a Riemannian manifold M with boundary does not exceed 1 2 dim M (dim M − 1). Then we classify such Riemannian manifolds M with boundary that the isometry groups I(M ) attain the preceding maximal dimension.…”

“…The metric g M in this case is always complete. 1), then there exist a number T ∈ (0, ∞] and a smooth function f : [0, T ) → (0, ∞) such that the metric tensor g M on M can be expressed by…”

“…In Section 2, we prove the fact that the two definitions of isometry coincide on Riemannian manifolds with boundary (see Proposition 2.1). It seems that Bagaev-Zhukova [1] did not mention this fact in their setting of Riemannian orbifolds. The idea of making reduction to the boundary in the proof of Proposition 2.1 will be used many times afterwards.…”

“…In this section we also show the above mentioned observation that the isometry group I(M ) of a Riemannian manifold M with boundary is a Lie transformation group of dimension at most 1 2 dim M (dim M − 1) (see Theorem 2.1). Although our proof of the observation is based on the idea of Proposition 2.1, to avoid the troublesome argument of point set topology, we also use the result in [1] that I(M ) has a Lie group structure. In Sections 3-5, we use the metric geometry and the theory of transformation group to prove Theorems 1.1-1.3.…”