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I have several times seen stated the fact that the complex projective space with the Fubini-Study metric is isotropic, but i can't seem to find the proof anywhere, can anyone suggest a book or answer with a proof?

Lee Mosher
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Zackury
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  • See for example here. The Fubini-Study metric on $\mathbf{CP}^n$ is isotropic, and the simply-connected space forms (the n-sphere, hyperbolic space, and $\mathbb {R} ^{n})$ are isotropic. One answer says "A perfect reference for isotropic manifolds is the book by Joe Wolf, Spaces of Constant Curvature." I agree. – Dietrich Burde Jun 08 '23 at 14:01
  • I was looking for a proof of the fact that you correctly stated, but i can't find it in the question you linked, i will check the book you suggested tho, thanks – Zackury Jun 08 '23 at 14:44
  • I couldn't find a proof in the book, also once again the question that you linked does not contain an actual dimostration of the fact. – Zackury Jun 09 '23 at 07:38
  • You couldn't find a proof in Joe Wolf's book? But it is there! Theorem $8.12.2$ shows that these spaces are two point homogeneous, and by Lemma $8.12.1$ they are hence isotropic. Did you really open this book? – Dietrich Burde Jun 09 '23 at 07:47
  • Thanks, i found it now there is no need to be rude – Zackury Jun 09 '23 at 14:41
  • You said "I couldn't find a proof in the book, also once again the question that you linked does not contain an actual dimostration of the fact". I find this equally "rude", implying that my reference is useless, although this isn't true. Even more, you might not even have looked into the book. That would be really rude then, to say it is not in there, without looking into it. But I trust you did, so all is settled now. – Dietrich Burde Jun 09 '23 at 14:51

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