In the linked paper, I am unable to understand how the expression (4) is arrived at??
To elaborate, the authors take the curvature operators of two isometric surfaces $M$ and $\overline{M}$ as $\boldsymbol{\kappa}$ and $\boldsymbol{\overline\kappa}$, and try to simplify an expression of the form
$$ E = \dfrac{\lambda}{1+v^2}\big[(1-v)Tr(\boldsymbol{\kappa}-\boldsymbol{\overline\kappa})^2 - v. Tr^2(\boldsymbol{\kappa}-\boldsymbol{\overline\kappa})\big]$$
where $Tr$ is the trace of the operator and $\lambda, v $ are constants.
Essentially the surface $M$ has been bent isometrically into $\overline{M}$ and the corresponding points on the surfaces are identified via the isometry. They say if $H = \frac{1}{2}(\kappa_1 + \kappa_2)$ is the mean curvature of M and similarly $\overline{H}$ of $\overline{M}$, then writing $D = \frac{1}{2}(\kappa_1 - \kappa_2)$ and respectively $\overline{D}$ for $\overline{M}$, we obtain
$$E = (H-\overline{H})^2 + \gamma(D - \overline{D})^2 +4D\overline{D}\sin^2{\theta},$$
where $\theta$ is the angle between the principal directions of $\boldsymbol{\kappa}$ and $\boldsymbol{\overline{\kappa}}$ and $\gamma = \dfrac{(1+v)}{(1-v)}$. What does this mean precisely??I am unable to figure out the exact definition of $\theta$ in terms of $\boldsymbol{\kappa}$ and $\boldsymbol{\overline{\kappa}}$. That would also help in understanding the derivation.