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Jun 27, 2017 at 3:10 comment added Igor Belegradek The isometry group is Lie group that acts properly and conversely, any proper effective action of a Lie group is isometric in some Riemannian metric. The former is proven e.g. in Kobayashi-Nomidzy textbook, vol I, Theorem I.4.7. The latter is due to Palais Theorem 4.3.1 in [On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961), 295–323.] See vmm.math.uci.edu/ExistenceOfSlices.pdf.
Jun 27, 2017 at 1:52 comment added Feng Hao If $G$ is compact, there is always a $G$-invariant metric over $M$, by integration over $G$. For General $G$, it should not be true
Jun 27, 2017 at 0:53 history edited user11881 CC BY-SA 3.0
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Jun 27, 2017 at 0:45 history asked user11881 CC BY-SA 3.0