Invariants and quasi-umbilicity of timelike surfaces in Minkowski space \(\mathbb R^{3,1}\) (Q423700)

The paper gives a characterization of time-like surfaces in Minkowski space. It is a continuation of the authors' study of the problem. In their previous work, the geometric invariants and principal configurations of space-like surfaces were presented [Proc. R. Soc. Edinb., Sect. A, Math. 140, No. 6, 1141--1160 (2010; Zbl 1205.53011)]. In the case of \(\mathbb R^4\), the geometry of immersed surfaces is characterized by classical invariants (the Gaussian and normal curvature, the norm of mean curvature and local convexity \(\Delta\)) and the curvature ellipses. The pseudo-Euclidean geometry of the Minkowski space yields the modification of the above set of invariants, e.g., instead of the curvature ellipse, one has to consider the curvature hyperbola. In Lorentz geometry two types of induced metrics of surfaces with signatures \((1,1)\) and \((2,0)\) appear. The authors consider them and characterize points of time-like surfaces introducing adequate invariants. Moreover at specific points, were the surface is quasi-umbilic, they find new invariant (not present for space-like surfaces).NEWLINENEWLINEThe paper consists of the following sections: Invariants of time-like surfaces; Quadratic maps from \(\mathbb R^{(1,1)}\) to \(\mathbb R^2\); Interpretation of the new invariant using Gauss map; Asymptotic and mean directionally curved directions.