Dear Professor Magda Toda,

B.Y. Chen [Mean curvature and shape operator of isometric immersions in real-space-forms, Glasgow Math. J. 38 (1996), 87–97] established the inequality (the generalization of classical Euler inequality)

\[K\leq \left\Vert H\right\Vert ^{2}+c\]   (K≤‖H‖²+c)

for any surface in a Riemannian space form M^{m}\left( c\right) . The equality holds identically if and only if the surface is a totally umbilical.

Hope it gives an useful idea to you. Best regards. Thank you.

Hi,

If the ambient space is Minkowski, then Null-scrolls also satisfy the given equality for c=0. It looks like they are only surfaces in E^3_1 with non-diagonalizable shape operator satisfying H^2=K. 

Professor Aydin, Benoit, and Turgay - many thanks  for your answers and information. All your answers are relevant and very useful to me, thanks. I was asking about the case when $ H^2 - K = a $ represents an arbitrary constant (for a surface immersed in Euclidean space). Meanwhile, another expert told me they are called quadratic Weingarten surfaces but there is no specialized name that he knows other than that. Whenever the mean and Gaussian curvatures (H and K) of a surface satisfy a quadratic polynomial equation, a surface is said to be quadratic Weingarten. It would be interesting if there existed any given names for special subcases. Of course, points where $H^2 = K$ are umbilics.... any special names for points that satisfy $H^2 - K = a$?

Dear Professor Magda,

I am sending two papers as documents. I hope, they are useful for you. 

Sincerely.

Dear Alev, thanks, the papers are interesting and I am reading them with pleasure. Still, I am interested in the most general case as written above. Many thanks in any case.

Thanks, Nikos, I was asking for a name for such surfaces if one is known.  Other than quadratic Weingarten....

Dear Professor Toda:

$H^2-K$ is known as the difference curvature since $H^2-K={(k_1-k_2)/2)}^2$, where $k_1$ and $k_2$ are the two principal curvatures of the surface {see [Chen, On the difference curvature of surfaces in euclidean space.  Math. J. Okayama Univ. 14 (1969), 153–157].)

For this reason, a surface satisfying $H^2-K=a$ is nothing but a surface with constant difference curvature.

Sincerely, 

Bang-Yen Chen

Many thanks!!

Just a small note for the benefit of other recent readers of this question :)

In their recent paper

Article A Note on the Class of Surfaces with Constant Skew Curvatures

Magdalena and Alexander referred to the surfaces in question as to those with constant skew curvature.

Thank you prof. Sergyeyev,

Yes, in our recent paper, we referred to the surfaces in question as constant skew curvature surfaces (CSkC surfaces)

in dif geometry of warped submanifolds of some kahler, cosymplektic manifolds

we can docollobradions...

This is one kind of CMF surface . For the CMS surface, can see the paper at" Differential Geometry and its Applications 39 (2015) 20–35"

Dear Professors Cenap Özel and Zhen Guo,

Thank you for your very interesting answers!!