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Let $(\mathbb CP^n,g_{FS})$ be the complex projective space equipped with the standard Fubini-Study metric.

What is the Riemannian isometry group of $(\mathbb CP^n,g_{FS})$? It seems to me that its identity component must be the projective unitary group.

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    $\begingroup$ That's correct. Moreover, the Riemannian isometry group, which has two components, is generated by the projective unitary group together with any anti-holomorphic isometry. $\endgroup$
    – Robert Bryant
    Commented Jun 24 at 18:44

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