Simple umbilic points on surfaces immersed in \(\mathbb{R}^4\) (Q1395983)

This paper involves a study of the possible configurations of the lines of principal curvatures around umbilic points on a surface \(f: M \to \mathbb R^4\) immersed in a 4-dimensional Euclidean space. As such the paper is a continuation of an earlier work co-authored by the authors \textit{C. Gutierrez} et al. [Bol. Soc. Bras. Mat., Nova Sér. 28, No. 2, 233--251 (1997; Zbl 0893.53004)]. A line of principal curvature is determined by a principal direction \(X\), which is a unit vector in the tangent space of the surface for which the length of \(\alpha(X,X)\) is extremal (\(\alpha\) denotes the second fundamental form). In isothermal coordinates \(z = u + iv\) the differential equation of the lines of curvature is \[ 4 a(u, v)(du^2 - dv^2)du\, dv + b(u, v)(du^4 - 6du^2 dv^2 + dv^4) = 0, \] where \(a\) and \(b\) are smooth real valued functions represented by the real and the imaginary parts of \(\langle \sigma , \sigma\rangle \), with \(\sigma = \alpha (\partial _z, \partial _z)\). Around an umbilic point \(p \in M\) with isothermal coordinates \((0, 0)\) one can find constants \(A\) and \(B\) and real-valued functions \(S(u, v)\) and \(R(u, v)\) such that \(a(u, v) = Au + Bv + S(u, v)\) and \(b(u, v) = v + R(u, v)\). The slope of a principal line approaching the origin (i.e. the umbilic point) is a real root of the separatrix polynomial \(g(s) = -sQ(s)\), where \(Q(s) = s^4 - 4Bs^3 - 2(3+ 2A)s^2 + 4Bs + 1 + 4A\). Around a simple umbilic point \(p\) exist the following possibilities for the nature of the roots of \(g\): (a) The separatrix polynomial has only simple roots, in which case \(p\) is a hyperbolic umbilic point; (b) \(g\) has a root of multiplicity two, in which case \(p\) is called an \(H_{34}\) umbilic point; (c) \(g\) has a root of multiplicity three, and \(p\) is called an \(\widetilde H_3\) umbilic point. In the above-mentioned paper it was shown that the hyperbolic umbilic points are locally topologically stable and that there are three different topological types \(H_3, H_4,\) and \(H_5\) of them. The other two possibilities (b) and (c) lead to unstable umbilic points and they are the main topic of disscusion in the present paper. All five types are characterized by different combinations of algebraic signs of the discriminant \(\Delta\) of \(Q\) and values of \(A\) and \(B\). Local behavior of the polynomial \(g(s)\) around the roots determines the bifurcation diagram of the umbilic point and the corresponding possible five phase portraits. In the last section of the paper, the authors obtain versal unfoldings of the unstable umbilic points \(H_{34}\) and \(\widetilde H_3\), which are presented as a family of Monge surfaces \(f_\lambda : \mathbb R^2 \to \mathbb R^4\) of the form \(f_\lambda (x, y) = (x, y, R_\lambda (x, y), S_\lambda (x, y))\), with explicit values for \(R_\lambda , S_\lambda \) given.