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By an argument using Liouville's theorem for conformal maps Conformal automorphism of $H^n$ it can be shown that every conformal automorphism of a hyperbolic manifold is an isometry.

Are there any other symmetric spaces (and thus the corresponding class of locally symmetric spaces) for which all conformal automorphisms are actually isometries?

For some symmetric spaces there are obviously non-isometric conformal automorphisms, e.g. an inversion in a hypersphere is a conformal transformation of $ E^n $ that is not an isometry.

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  • $\begingroup$ For other hyperbolic spaces, you have an isomorphism between their isometry group and the natural automorphism group of the parabolic geometry of their sphere at infinity (conformal of $H^n$, CR for $CH^n$, etc.). I don't know if they correspond to the conformal group of said hyperbolic space (but I suspect the answer is no) $\endgroup$
    – Didier
    Sep 11 at 16:50

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It is proven in Corollary 1 of

Nagano, Tadashi, The conformal transformation on a space with parallel Ricci tensor, J. Math. Soc. Japan 11, 10-14 (1959). ZBL0089.17201.

that if $M$ is a connected locally-symmetric space not isometric to $S^n$ or ${\mathbb R}^n$ (with, respectively, metrics of positive and zero constant sectional curvature), then the group of conformal transformations of $M$ equals the group of isometries of $M$.

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