A curved line in $\mathbb R^3 $ has properties of curvature and torsion, and, on an $ \mathbb R^2 $ possesses surface scalar properties of normal curvature and geodesic torsion $ (\kappa_n, \tau_g).$

In isometric mappings product of extreme $\kappa_n's$ remains constant by Egregium theorem.

Euler normal curvatures are from tangent plane $ (\dfrac{dT}{ds}= \kappa_n N ): $

$$ \kappa_n= \kappa_1 \cos^2 \psi+\kappa_2 \sin^2 \psi $$

What about $\tau_g ?$ Is there any theorem corresponding to curvature dealing with torsions as well? Like in normal plane ( since $ \dfrac{dB}{ds}= -\tau_g N ):$

$$ \tau_g = \tau_1 \cos^2\phi + \tau_2 \sin ^2\phi ?$$

where $\phi$ is angle between principal binormal direction and reference binormal direction.

And what about the product of extreme $\tau_g's ? $ Does a corresponding scalar $ "T " $ also remain constant in isometry?

The question is about conditions for existence of an isometric twisting invariant $"T"$ gleaned from this geometrical analogue just as Gauss curvature $K$ is a bending invariant.

The statement needs to be improved to remove vagueness of expressing torsion. I request your help to refine the question and point to an answer to the extent it makes sense in this context. And please suggest references also.

I have tried to represent ( a rough sketch.. shall improve it) the two parameters in bipolar coordinates together. Motivation for bipolar plot representation of curvature/torsion tensor view is.. to work on a prospect to build premises that if $K$ invariance is true then $T$ invariance must consequently be also true...and a twist invariant $T$ that rank with $K$ would be found/identified.

EDIT 1:

I mean we should be able to formulate for a surface of revolution of constant torsion product T.

T=K in magnitude as geometric mean of segment lengths as shown in rough sketch.. but its definition is drastically different. Can we have a formulation (and views if possible) of any example of a surface of constant P with parametrization obtained by integration of an ODE with arbitrary constants in two or three twisted stages where individual principal geodesic torsions multiply to same product P? The answer to such question for constant K is well known.

A metric in normal $(B-N)$ plane has to be found.

I wonder if topic of Spinor Bundles is relavant here.