[Math] Isometric but differently shaped surfaces in $\mathbb{R}^3$

differential-geometry

We have the following chain of inclusions for surfaces in $\mathbb{R}^3$ $M_1,M_2$:

  1.      $M_1,M_2$ have the same shape, i.e. are related by an ambient isometry
    ⇆ $M_1,M_2$'s first and
    second fundamental forms agree

  2. → $M_1,M_2$ are isometric
    ⇆ $M_1,M_2$'s first fundamental forms agree

  3. → $M_1,M_2$ have the same Gaussian curvatures

  4. → $M_1,M_2$ have the same genus (for closed surfaces)

In a more catchy way: shape → metric → curvature → genus

I know the standard examples of isometric but differently shaped surfaces in $\mathbb{R}^3$: plane, cone, cylinder.

I am looking for (other) examples of

  • isometric but differently shaped surfaces (preferrably closed ones)

  • non-isometric surfaces with the same curvature

I assume there are no differently shaped surfaces isometric to the sphere, are there?

But what about other convex surfaces (with strictly positive but not constant curvature)? Or arbitrary surfaces homeomorphic to the sphere? Or to the torus?

(A picture gallery would be highly welcome, because I really would like to see two such (non-)isometric surfaces.)

Best Answer

There is a continuous, isometric deformation between a catenoid and a helicoid.

A parametrization of such a deformation is given by the system $$\begin{align} x(u,v) &= \cos \theta \,\sinh v \,\sin u + \sin \theta \,\cosh v \,\cos u\\ y(u,v) &= -\cos \theta \,\sinh v \,\cos u + \sin \theta \,\cosh v \,\sin u\\ z(u,v) &= u \cos \theta + v \sin \theta \, \end{align}$$

for $(u,v) \in (-\pi, \pi] \times (-\infty, \infty)$, with deformation parameter $-\pi < \theta \le \pi$, where $\theta = \pi$ corresponds to a right-handed helicoid, $\theta = \pm \pi / 2$ corresponds to a catenoid, and $\theta = 0$ corresponds to a left-handed helicoid.

In fact, there are lots of such families of isometric minimal surfaces.

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