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It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.

I discovered lately that in dimension $d>2$, conformal harmonic maps must be scaled isometries (The conformal factor is constant).

This is a "rigidity phenomena"- in stark contrast to the $2$D case, in higher dimensions the set of conformal and harmonic maps is very "small".

I am quite sure this should be already known, but couldn't find a reference. Any help?

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  • $\begingroup$ I saw it stated in the literature (but without a reference). $\endgroup$ Apr 19, 2017 at 22:43
  • $\begingroup$ Thanks, do you remember where did you see it? (although as you said, it probably won't be of much help, since there is no reference). $\endgroup$ Apr 20, 2017 at 8:16

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After posting in MO, it turns out that:

Indeed, the result can be found in the book Harmonic morphisms between Riemannian manifolds, by Paul Baird, John C. Wood.

The relevant statement is Corollary 3.5.2.

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