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I have a rather elementary question.

I would like to know what are the surfaces of $\mathbb{R}^3$ which are globally preserved by the action of a linear or affine map in a non trivial way. This question is voluntarily vague.

Obvious example of such surfaces are linear planes and the level sets of quadratic form. So my questions unfold the following way:

  1. Do 'exotic' surfaces invariant by an element $A \in \mathrm{GL}(3,\mathbb{R})$ or $\mathrm{Aff}(\mathbb{R}^3)$ exist? (in the sense that it is neither a plane nor a level set of a quadratic form)

  2. Are there example of such surfaces which are invariant by a Lie subgroup of $\mathrm{GL}(3,\mathbb{R})$ or $\mathrm{Aff}(\mathbb{R}^3)$?

  3. If such a surface $\Sigma$ exists, is the action of an element $A \in \mathrm{GL}(3,\mathbb{R})$ or $\mathrm{Aff}(\mathbb{R}^3)$ determined by its value at a point $p$ and by $A_{|T_p\Sigma}$?

  4. Can one list all such surfaces?

Immersed or degenerated surfaces would be interesting to me as well. Any reference is welcome! Thanks :)

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    $\begingroup$ Surfaces of revolution give an example for questions 1. and 2. Also, the surface $(x^2+y^2)z^2 = 1$ is invariant under a two-dimensional subgroup generated by rotations around the $z$-axis and $(x,y,z) \mapsto (\lambda x, \lambda y, \lambda^{-1} z)$. $\endgroup$
    – Ivan Izmestiev
    Commented May 8, 2017 at 15:08

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For immersed surfaces, take your favorite surface $S$ and your favorite affine map $\gamma$ and look at the $\Gamma$-orbit of $S$ (where $\Gamma = \langle \gamma \rangle)$.In particular, the case where $\gamma$ is an involution gives you a boat-load of examples. Otherwise, take an arbitrary curve and look at its image under a one-dimensional Lie subgroup of $Aff(3).$ An embedded example is the orbit by translations in the $z$ direction of an arbitrary curve in the $xy$-plane. Otherwise, take an orbit of a discrete point set under a two dimensional subgroup...

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  • $\begingroup$ You might be more generous and allow simultaneous rotation and translation of plane curves, to produce helicoids as the easiest example. $\endgroup$
    – Sebastian Goette
    Commented May 9, 2017 at 15:24
  • $\begingroup$ @SebastianGoette True enough! $\endgroup$
    – Igor Rivin
    Commented May 9, 2017 at 18:19

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