For shallow geometry of a surface using rectangular we have from classical St. Venant relation

$$  \dfrac{\partial ^2{\epsilon_x}}{{\partial {y^2}} }+ \dfrac{\partial ^2{\epsilon_y}}{{\partial {x^2}} }- \dfrac{\partial ^2{\gamma_{xy}}} {\partial {x}\partial {y} }  $$

that links direct and shear strains for compatibility and is known to represent Gauss curvature derived by Von Karman.  It is an isometric invariant that can derived from first fundamental form of surface theory.

But what about the other geometrical invant of St. Venant,

$$  \dfrac{\partial }{{\partial {x}}}  \big[ \dfrac{\partial {\gamma_{zx}}} {\partial {y} }+  \dfrac{\partial {\gamma_{xy}}} {\partial {z} } -\dfrac{\partial {\gamma_{yz}}} {\partial {x} } \big] -  2\dfrac{\partial ^2{\epsilon_x}}{{\partial {y} \partial {z}  } } ?$$

Is it  of isometric/ topological invariants derived from first and fundamental forms? If so, how is it done?

Regards