  Here are studied pairs of transversal foliations with singularities, defined
on the Elliptic region (where the Gaussian curvature $\mathcal K$ is positive)
of an oriented surface immersed in $\mathbb R^3$. The leaves of the foliations
are the lines of geometric mean curvature, along which the normal curvature is
given by $\sqrt {\mathcal K}$, which is the geometric mean curvature of the
principal curvatures $ k_1, k_2$ of the immersion. The singularities of the
foliations are the umbilic points and parabolic curves}, where $ k_1 = k_2$ and
${\mathcal K} = 0$, respectively. Here are determined the structurally stable
patterns of geometric mean curvature lines near the umbilic points, parabolic
curves and geometric mean curvature cycles, the periodic leaves of the
foliations. The genericity of these patterns is established. This provides the
three essential local ingredients to establish sufficient conditions, likely to
be also necessary, for Geometric Mean Curvature Structural Stability. This
study, outlined at the end of the paper, is a natural analog and complement for
the Arithmetic Mean Curvature and Asymptotic Structural Stability of immersed
surfaces studied previously by the authors.
