generalisation of umbilic surfaces

Question

It is well known that if you have a complete surface in $\mathbb{R}^3$ with umbilic points, that is to say $k_1=k_2$ everywhere, where $k_1$ and $k_2$ are the principal curvatures, that is to say the eigenvalues of the second fundamental form, then is a plane or a sphere.

Is something known for $k_1-k_2=constant$? Of course the cylinder provides an example, but is there some example with $k_1$ non constant?

Finally I wonder if it may be easier to have example in higher codimension? For instance for the case of surfaces in $\mathbb{R}^d$.

Answer 1

In general, surfaces in $\mathbb{E}^3$ for which the principal curvatures satisfy a given functional relation $F(\kappa_1,\kappa_2)=0$ are said to be Weingarten surfaces (of type $F$), and the condition for a graph $z = f(x,y)$ to be a Weingarten surface of type $F$ is a single second order PDE for the function $f(x,y)$. The general theory tells you that, locally, the Weingarten surfaces of type $F$ depend on 2 'arbitrary' functions of one variable. In particular, there are lots of such surfaces locally.

For example, a surface of revolution of the form $z = h(x^2+y^2)$ will satisfy $\kappa_1-\kappa_2 = 1$ if and only if $h(r)$ for $r>0$ satisfies a certain second order ODE that is not difficult to write down. So, yes, there are other solutions besides cylinders, even surfaces of rotation.

More generally, if you specify a real analytic connected curve in $\mathbb{E}^3$ and a real-analytic normal vector field along it, then, subject to some open conditions on the curve and the vector field, there will usually be a unique real analytic connected surface in space that contains the curve, has the specified normal along the curve, and satisifes $\kappa_1-\kappa_2 = 1$.

Answer 2

..a small addition to Professor Bryant's answer

For this class of surfaces, where $k1-k2 = constant$, we have introduced the name "Costant Skew Curvature Surfaces" (CSkC-surfaces) and we have studied an aspect concerning the Bonnet-surfaces.

It is well know that the famous question that Bonnet asked was: "When does there exist an isometric embedding $x:M^2 \rightarrow R^3$ such that the mean curvature function of the immersion is $H$?"

Our work was born from the question: Can a surface be CSkC and Bonnet at the same time, and, if that is the case, what does it represent?

We showed that the CSkC-surfaces with principal curvatures ($k_1$ and $k_2$) nonconstant cannot contain any Bonnet-surfaces, so if and only if $k_1$ and $k_2$ are both constant the class of CSkC-surfaces can admit Bonnet-surfaces.

This means that the only CSkC-surfaces for which exists a nontrivial isometric deformation preserving the mean curvature $H$ are (patch of) a circular cylinders.

see (Alexander Pigazzini, Magdalena Toda): https://projecteuclid.org/euclid.jgsp/1518577293

Another aspect of the CSkC-surfaces is that if we setting $k_1-k_2=constant$ renders the shape equation for an elastic membrane equivalent to the Schrodinger equation for a particle on the same surface.. then the same equations have the same solutions...

see (Victor Atanasov, Rossen Dandoloff): https://iopscience.iop.org/article/10.1088/0143-0807/38/1/015405/pdf

By relating the two works we can for example say that:

"if you want to look for an isometric deformation that preserve this relation (shape equation equivalent to the Schrodinger equation) then you can get the result only if the surface is (patch of) circular cylinder, and then search through its isometric deformations".