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Let $(M^2,g,\boldsymbol{\xi})$ be a two-dimensional Riemannian manifold endowed with a Riemannian metric $g$ (strictly positive signature) and a Killing field $\boldsymbol{\xi}$. Will there be an isometric embedding into $\mathbb{R}^3$ such that it is symmetric under rotations about some straight line ?

DanielKatzner
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    Do you assume compactness of the surface? Embedding in what space? What kind of an embedding? (Presumably, isometric?) What did you try to answer your own question? – Moishe Kohan Mar 17 '22 at 13:10
  • @MoisheKohan Thanks. Yes, isometric embedding in 3 dimensional Euclidean space. I would be interesting in both compact and non-compact cases. I guess the answer is not because there may be Killing fields that act non-trivially, but for the moment I cannot figure out a counterexample. – DanielKatzner Mar 17 '22 at 14:25
  • Not a duplicate, but close enough to resolve the question in the negative: https://math.stackexchange.com/questions/2370393/does-this-first-fundamental-form-imply-a-surface-of-revolution – Andrew D. Hwang Mar 17 '22 at 15:39
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    Then there are all kinds of obstructions to the existence of a rotationally symmetric isometric embedding. Start from the fact that hyperbolic plane has no isometric immersions in $E^3$, regardless of symmetry. Flat torus does not admit isometric immersions if you care about compact surfaces. – Moishe Kohan Mar 17 '22 at 16:48

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