A conjecture in relation to Loewner's conjecture
Let \(f\) be a smooth function of two variables \(x\) and \(y\) on a domain \(D\) of \(\mathbb R^2\). For a positive integer \(n\in\mathbb N\), let \(d^nf\) be a symmetric tensor field of type \((0,n)\) defined by \[ d^nf:=\sum_{i=0}^n\binom ni \frac{\partial^nf}{\partial x^{n-i}\partial y^i}dx^{n-i}dy^i \] and \(\tilde\mathcal D_{d^nf}\) be a finitely many-valued one-dimensional distribution obtained from \(d^nf\) (e.g., \(\tilde{\mathcal D}_{d^1f}\) is the one dimensional distribution defined by the gradient vector field of \(f\); \(\tilde\mathcal D_{d^2f}\) consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of the Hessian of \(f\)). In the paper, the behavior of \(\tilde\mathcal D_{d^nf}\) around an isolated umbilical point of \(d^nf\) is studied. In particular, the index of an isolated umbilical point with respect to \(\tilde\mathcal D_{d^nf}\) is defined and treated. It is shown that the index is a rational number and not always represented as the half of an integer. A conjecture which asserts that the index of an isolated umbilical point with respect to \(\tilde\mathcal D_{d^nf}\) is not greater than one is investigated. It is shown that for \(n\in\{1,2\}\) (respectively, \(n\geq 3\)), this conjecture is equivalent to (respectively, distinct from) Loewner's conjecture. The former conjecture is affirmatively solved in the case where \(f\) is a homogeneous polynomial. In addition, this conjecture is studied in the case where \(f\) is a real-analytic function.
many-valued one-dimensional distribution
scientific article; zbMATH DE number 2164233