There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere.

What is known about isometric deformations of the sphere, when the smoothness condition is relaxed?

First, we need some notion of isometry for surfaces with singularities. Intuitively, if I take a piece of paper and crease it, the creased paper is still isometric in a weaker sense to the original flat sheet. More generally, for any topological surface embedded in $\mathbb{R^3}$, shortest distance between two points can still be well-defined as the (possibly infinite) infimum of the arc lengths of all curves on the surface joining the two points. An isometry is then any homeomorphism that preserves this distance.

Is anything known about the isometries of the sphere, when the surface is no required to remain $C^2$? Perhaps it's easier to look at isometric deformations -- continuous functions $f(s):\mathbb{R}\times\mathbb{R}^3\to\mathbb{R}^3$ with $f(0)$ the identity, and $f(s)$ an isometry of the unit sphere for all $s$.

There are some obvious isometric deformations of the sphere, for instance, cutting the sphere along a latitude, inverting one piece, and gluing it back on: $$f(s): (x,y,z)\mapsto (x,y, s-1+|z+1-s|)$$ and of course this deformation can be done at any point on the sphere, not just the south pole, and at different speeds, generating a large family of isometric deformations of the sphere.

Are there any others? Have the isometries been classified?


Rigidity persists a little below $C^2$, namely in the class $C^{1,\alpha}$ with $\alpha>2/3$. It breaks down for small Hölder exponents $\alpha$ in a rather strong way: any continuous embedding can be uniformly approximated by $C^{1,\alpha}$ isometric embeddings. (Key term: h-principle). This is a stronger form of Nash's $C^1$ isometric embedding theorem, which was the precursor to the h-principle. See h-Principle and Rigidity for $C^{1,\alpha}$ Isometric Embeddings by Sergio Conti, Camillo De Lellis, and László Székelyhidi Jr.

So yes, the set of isometric deformation is way too large to admit classification.


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