Generalizing results of J. M. Oliver [J. Lond. Math. Soc., II. Ser. 83, No. 3, 755–767 (2011; Zbl 1216.53008)] from surfaces in \(\mathbb{R}^3\) to two-dimensional surfaces in \(\mathbb{R}^4\), the author studies aspects of projective invariance and singularity of characteristic curves and a couple of other curves of differential geometric interest at \(P_3(c)\)-points. These are parabolic points in which the characteristic curve (and also the other curves) are tangent to the curve of parabolic points.
As in the case of surfaces in \(\mathbb{R}^3\), characteristic curves, asymptotic curves and principal curves are obtained by integrating directions given by binary differential equations \(Adx^2 + 2Bdx\,dy + Cdy^2 = 0\). Regarding the coefficients \(A\), \(B\), \(C\) as homogeneous coordinates in the real projective plane, asymptotic, principal and characteristic directions form an auto-polar triangle of the surface’s curvature ellipse.
The possibility for this projective interpretation is at the core of the projective invariance of the 2-jet of the characteristic inflection curve (the locus of geodesic inflection points) and the topological type of the characteristic curves’ singularity at a generic \(P_3(c)\)-point. All possible configuration of the special curves considered here at such a point can be classified in terms of cross-ratios of (some of) their directions.