I already asked thes question here, but received no answer. I was reading this interesting article by Givental

Givental, Alexander. "The Pythagorean theorem: What is it about?" Amer. Math. Monthly 113 (2006), no. 3, 261–265

And my attention was caught by this passage:

"This explains why the Pythagorean theorem is a genuinely Euclidean phenomenon (and not only in the historical sense of the word): among all Riemannian metrics of constant curvature only the Euclidean one admits nontrivial conformal isometries."

So that's the question: there exist Riemannian metrics of non constant curvature admitting nontrivial conformal isometries? If so, the Pythagorean theorem holds on them?

I already asked the same question here, but received no answer. I was reading this interesting article by Givental

Givental, Alexander. "The Pythagorean theorem: What is it about?" Amer. Math. Monthly 113 (2006), no. 3, 261–265

And my attention was caught by this passage:

"This explains why the Pythagorean theorem is a genuinely Euclidean phenomenon (and not only in the historical sense of the word): among all Riemannian metrics of constant curvature only the Euclidean one admits nontrivial conformal isometries."

So that's the question: there exist Riemannian metrics of non constant curvature admitting nontrivial conformal isometries? If so, the Pythagorean theorem holds on them?