Pairs of geometric foliations of regular and singular surfaces

We examine some generic features of surfaces in the Euclidean 3-space $\mathbb{R}^3$ related to the Gauss map on the surface. We consider these features on smooth surfaces and on singular surfaces with a cross-cap singularity. We study some symmetries between two classical pairs of foliations defined on smooth surfaces in $\mathbb{R}^3$: the asymptotic curves and the characteristic curves (called harmonic mean curvature lines in \cite{garciasotomayorharmonic}). The asymptotic curves exist in hyperbolic regions of surfaces and have been well studied. The characteristic curves are in certain ways the analogy of the asymptotic curves in elliptic regions. In this thesis we extend this analogy. We use We produce results on the characteristic curves mirroring those of Uribe-Vargas (\cite{uribevargas}) on the asymptotic curves. By considering cross-ratios of Legendrian lines in the manifold of contact elements to the surface we show that certain properties of the characteristic curves are invariant under projective transformations, and examine their behaviour at cusps of Gauss. We establish an analogy of the Beltrami-Enepper Theorem, which allows us to distinguish between the two characteristic foliations in a natural geometric way. We show that the local properties of characteristic curves may be used to prove certain global results concerning the elliptic regions of smooth surfaces. Motivated by the study of the asymptotic, principal and characteristic curves on surfaces in $\mathbb{R}^3$, we construct a natural one-to-one correspondence between the set of non-degenerate binary differential equations (BDEs) and linear involutions on the real projective line. We show that one may construct pairs of BDEs that have various symmetric properties using a single involution on $\mathbb{R}P^1$. We study the folded singularities of BDEs, and associate an affine invariant to such points. We show that one may associate a complex parameter to folded singularities that determines the relative positions of various curves of interest. We show that the BDEs asymptotic, characteristic, and principal curves are related to other quadratic forms on surfaces. These include the BDE that defines the lines of arithmetic mean curvature which are studied in \cite{garciasotomayorarith}, and the third fundamental form of the surface. We define a new pair of foliations of a surface which we label the minimal orthogonal spherical image (MOSI) curves which are the integral curves of those tangent directions to a surface that have orthogonal images under the Gauss map, and are inclined at an extremal angle. We establish the configurations of the MOSI curves in a neighbourhood of umbilic points, parabolic points and cusps of Gauss. We construct natural 1-parameter families of BDEs that interpolate between the BDEs we have studied, and establish relationships between these families. We exhibit the existence of a curve of points of zero torsion of the characteristic curves, and a curve of points where the tangent plane to the surface is the osculating plane of a characteristic curve. We determine the behaviour of these curves near cusps of Gauss and umbilic points. We study BDEs with coefficients that vanish simultaneously at an isolated point and with discriminant having an $A_2$-singularity at that point. We show that such BDEs can be grouped into three distinct types, and study the differences between these types in terms of their codimension and the linear parts of their coefficients. We establish the topological configurations of the solution curves in each case with codimension $\leq4$. We study the asymptotic and characteristic curves in the neighbourhood of a parabolic cross-cap, that is, on a singular surface with a cross-cap singularity with a parabolic set having a cusp singularity at the singular point. We obtain the topological configurations of these foliations both in the domain of a parametrisation of such a surface, and on the surface itself. We construct a natural one-parameter family of surfaces with cross-cap singularities in which the parabolic cross-cap is the transition from a hyperbolic cross-cap to an elliptic cross-cap. We study the bifurcations of the asymptotic and characteristic curves in this family.