# With respect to whom are you critical?

## TL;DR

This paper investigates the set of points for which a given point on a Riemannian surface is critical, establishing bounds on its size and properties related to the surface's topology and metric genericity.

## Contribution

It extends previous work by providing bounds on the number of critical points related to a given point on orientable surfaces, including generic metric properties and topological implications.

## Key findings

- Existence of an open dense set of metrics with an odd number of critical points for a given point.
- Upper bounds on the number of critical points: 5 for the torus, and 8g-5 for genus g surfaces.
- Properties of points at maximal distance on the surface are discussed.

## Abstract

For any compact Riemannian surface $S$ and any point $y$ in $S$, $Q_y^{-1}$ denotes the set of all points in $S$, for which $y$ is a critical point. We proved \cite{BIVZ} together with Imre B\'ar\'any that card$Q_y^{-1} \geq 1$, and that equality for all $y\in S$ characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface $S$ and any point $y \in S$, the following two main results. There exist an open and dense set of Riemannian metrics $g$ on $S$ for which $y$ is critical with respect to an odd number of points in $S$, and this is sharp. Card$Q_y^{-1} \leq 5$ for the torus and card$Q_y^{-1} \leq 8g-5$ if the genus $g$ of $S$ is at least $2$. Properties involving points at globally maximal distance on $S$ are eventually presented.