Surfaces of Constant Mean Curvature CMC and constant Gauss Curvature Product $K$ are well-known in differential geometry of surfaces of revolution. Denote half sum and half difference curvatures as

$$ k_1+ k_2 = 2 H_s; \, k_1- k_2 = 2 \, H_d. $$

**Significance of $H_d$**

Now I assume that $H_d$ may as well be of equal interest with fundamental importance. The assumption is motivated by its representation in * Mohr's Circle* radius of curvature for stress, shell curvatures, moment of inertia (in Figure) among other such tensors. As is known, material stress failure theories in structural mechanics operate on stress/strain and other tensoral

*differences*as

*entities. It occurs as an important curvature invariant in the equation of Mohr's Circle :*

**shear**$$\boxed{ (k_n-H_s)^2 +\tau_g^2= H_d^2} $$

I made a search in some textbooks that could be accessed. I was not lucky in finding references for the principal curvature difference profiles. Also the surface cannot be viewed as a particular case of CMC surfaces due to its sign change of curvature and so it is distinctly different. The calculation has a different expression and plots differently.

If $\phi$ is slope of tangent to meridian, primes on meridian arc

**Const $H=H_s$ Mean curvature Delaunay meridians**

$$\phi^{'}+\frac{\cos \phi}{r} =2 H_s$$

With initial radius $r=r_1$ at $\phi=0$

$$ \cos \phi = \frac{H_s (r^2-r_1^2)+r_1}{r}$$

$$(-k_1,k_2)= (H_s(r^2+r_1^2)-r_1,\, H_s(r^2-r_1^2)+r_1)\,$$

**Const $H_d$ Difference curvature meridians**

**First Order ODE:**

$$-\phi^{'}+\frac{\cos \phi}{r} = 2 H_d \tag1$$

First order ODE has the disadvantage of numerically computing indefinitely below points of singularity on $r=0$ axis so there appear multiple spindle meridian profiles below this symmetry axis.

**Second Order ODE:**

$$\phi^{''}= 2 H_d \tan\phi\, (\phi^{'} +2 H_d) \tag2 $$

$$ \cos \phi = \frac{r}{r_1}+2\, H_d\, r\, log \,\frac{r}{r_1}\tag3 $$

$$(k_1,k_2)=(\frac{1}{r_1}+2H_d(1+log \,\frac{r}{r_1}),\frac{1}{r_1}+2H_d \,log \,\frac{r}{r_1} ) \tag4 $$

$$ @\,r=0,\phi \rightarrow \pi/2$$

Second order ODE has the advantage of stopping computation at point of singularity so there appears no profile below symmetry line $r= 0.$

Shown below are profiles of constant difference $H_d$ of principal curvatures.

Three distinct shapes occur.

*Progressive loops* $ H_d <-0.5$ above $r=1$ ;

Two types below $r=1$

*Ovaloids* between cylinder and sphere $ 0>H_d>-0.5;$ and,

*Profiles with Inflection Point* for $ H_d >0 $ occurring at $ r=r_1e^{-(1+1/(2 r_1H_d))}. $

All profiles meet the axis of symmetry normally, however these are * not* umbilical points.

Thanks in advance for your comments and for any references available on the topic.

iswell-defined, it's $4(H^2-K)$; setting it equal to a constant is reasonable. There are many such surfaces (locally), most of which are not surfaces of revolution. It's true that such surfaces cannot have umbilic points, and any compact surface with $H^2-K>0$ being constant must be either a torus or a Klein bottle (if such exist, which is not obvious). The PDE that describes such surfaces is hyperbolic, though, so elliptic methods won't generally be of much help, and the few examples with symmetry won't tell you much. $\endgroup$1more comment