On the isometry group of $$RCD^*(K,N)$$ R C D ∗ ( K , N ) -spaces

Abstract

We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, RCD * (K, N ), is a Lie group. We obtain an optimal upper bound on the dimension of this group, and classify the spaces where this maximal dimension is attained.

The Isometry Group of an RCD ∗ Space is Lie

“…During the completion of this manuscript Guijarro and Santos-Rodríguez proved in [17,Theorem 1] that ISO m (X) is a Lie group for RCD * -spaces, compare with Corollary 1.2 above. The proof of Theorem 1 of [17] and that of our Theorem 1.1 both rely on the aforementioned result due to Gleason and Yamabe, however the approaches to the problem are different.…”

“…The proof of Theorem 1 of [17] and that of our Theorem 1.1 both rely on the aforementioned result due to Gleason and Yamabe, however the approaches to the problem are different. Guijarro and Santos-Rodríguez adapted the approach of [12] in which the existence of a splitting theorem is indispensable, thus only RCD * -spaces are contemplated.…”

Invariant Measures and Lower Ricci Curvature Bounds

“…In [24], and indenpendently in [37], it was shown that the isometry group of an RCD * (K, N ) space is a Lie group. Therefore it makes sense now to study properties of Lie group actions on m.m.s.…”

“…A priori one would assume that the metric structure and the measure have no relationship at all; however we have the following result. The following was obtained in [24], and independently by Sosa [37]. Theorem 2.19.…”

“…By [35] we know that the action is proper. Then the isotropy groups are always compact, the quotient space X/G is geodesic and we can lift geodesics (see [24] for details). This allows us to follow the same argument as in [17] and obtain the result.…”