For shallow geometry of surface in Monge form using rectangular $z=f(x,y)$ we have from classical St. Venant relation

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

that is known to represent Gauss curvature as derived by . It links direct and shear strains for compatibilty.

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

But what is the other geometrical invant of St. Venant?