The following question is related to this previous question, Canonical immersion of the double torus:

Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed with a metric of constant curvature K = -1 into a Hilbert space of finite dimension?

  • $\begingroup$ Isometric embedding into where? $\endgroup$
    – Deane Yang
    Jul 30, 2020 at 23:23
  • $\begingroup$ Into a Hilbert space of finite dimension. $\endgroup$ Jul 31, 2020 at 2:27
  • $\begingroup$ Or, to be more concrete, into $\mathbb{R}^n$ for some $n > 0$? $\endgroup$
    – Deane Yang
    Jul 31, 2020 at 2:51
  • $\begingroup$ Into an R^n with a fixed inner product. $\endgroup$ Jul 31, 2020 at 19:12
  • $\begingroup$ I’m pretty sure none is known, and it’s an open question whether any such “explicit” embedding or evening just immersion exists. $\endgroup$
    – Deane Yang
    Jul 31, 2020 at 22:11


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