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YCor
YCor
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Canonical Immersionimmersion of the Double Torusdouble torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the flat metric in the torus, and then, from the mathematical point of view, this immersion is highly preferred to the standard 'doughnut' one into $\mathbb{R}^3$.

I have always wondered whether there exists something similar for higher genus surfaces, for instance a genus 2 surface -– double torus. I imagine that, since the hyperbolic plane is the space used as universal cover for the double torus with constant curvature $-1$, one should search some smooth functions from the hyperbolic plane (say, the Poincaré disc in $\mathbb{C}$) to a higher Euclidean or Hiperbolichyperbolic space which are periodic respect to the action of some Fuchsian group. Having these functions, the point is that the immersion induces the $-1$ constant curvature.

Of course, there must be some such embedding, according to the theorems of isometric embeddability. But the aim of this question is to ask whether there exist some nice and natural functions which do the job as in the case of the torus.

Any suggestion is welcome.

Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the flat metric in the torus, and then, from the mathematical point of view, this immersion is highly preferred to the standard 'doughnut' one into $\mathbb{R}^3$.

I have always wondered whether there exists something similar for higher genus surfaces, for instance a genus 2 surface - double torus. I imagine that, since the hyperbolic plane is the space used as universal cover for the double torus with constant curvature $-1$, one should search some smooth functions from the hyperbolic plane (say, the Poincaré disc in $\mathbb{C}$) to a higher Euclidean or Hiperbolic space which are periodic respect to the action of some Fuchsian group. Having these functions, the point is that the immersion induces the $-1$ constant curvature.

Of course, there must be some such embedding, according to the theorems of isometric embeddability. But the aim of this question is to ask whether there exist some nice and natural functions which do the job as in the case of the torus.

Any suggestion is welcome.

Canonical immersion of the double torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the flat metric in the torus, and then, from the mathematical point of view, this immersion is highly preferred to the standard 'doughnut' one into $\mathbb{R}^3$.

I have always wondered whether there exists something similar for higher genus surfaces, for instance a genus 2 surface – double torus. I imagine that, since the hyperbolic plane is the space used as universal cover for the double torus with constant curvature $-1$, one should search some smooth functions from the hyperbolic plane (say, the Poincaré disc in $\mathbb{C}$) to a higher Euclidean or hyperbolic space which are periodic respect to the action of some Fuchsian group. Having these functions, the point is that the immersion induces the $-1$ constant curvature.

Of course, there must be some such embedding, according to the theorems of isometric embeddability. But the aim of this question is to ask whether there exist some nice and natural functions which do the job as in the case of the torus.

Any suggestion is welcome.

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Jjm
Jjm
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Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the flat metric in the torus, and then, from the mathematical point of view, this immersion is highly preferred to the standard 'doughnut' one into $\mathbb{R}^3$.

I have always wondered whether there exists something similar for higher genus surfaces, for instance a genus 2 surface - double torus. I imagine that, since the hyperbolic plane is the space used as universal cover for the double torus with constant curvature $-1$, one should search some smooth functions from the hyperbolic plane (say, the Poincaré disc in $\mathbb{C}$) to a higher Euclidean or Hiperbolic space which are periodic respect to the action of some Fuchsian group. Having these functions, the point is that the immersion induces the $-1$ constant curvature.

Of course, there must be some such embedding, according to the theorems of isometric embeddability. But the aim of this question is to ask whether there exist some nice and natural functions which do the job as in the case of the torus.

Any suggestion is welcome.