  Consider oriented surfaces immersed in $\mathbb R^3.$ Associated to them,
here are studied pairs of transversal foliations with singularities, defined on
the Elliptic region, where the Gaussian curvature $\mathcal K$, given by the
product of the principal curvatures $k_1, k_2$ is positive. The leaves of the
foliations are the lines of harmonic mean curvature, also called characteristic
or diagonal lines, along which the normal curvature of the immersion is given
by ${\mathcal K}/{\mathcal H}$, where $ {\mathcal H}=({k_1}+k_2)/2$ is the
arithmetic mean curvature. That is, ${\mathcal K}/{\mathcal H}=((1/{k_1} +
1/{k_2})/2)^{-1}$ is the harmonic mean 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 harmonic mean curvature
lines near the umbilic points, parabolic curves and harmonic 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 Harmonic Mean
Curvature Structural Stability of immersed surfaces. This study, outlined
towards the end of the paper, is a natural analog and complement for that
carried out previously by the authors for the Arithmetic Mean Curvature and the
Asymptotic Structural Stability of immersed surfaces.
