Remarks on Brouwer Fixed Point Theorem for Some Surfaces in $\mathbb{R}^3$

Abstract

Let $X$ be a surface in $\mathbb{R}^3.$ A subset $E$ of $X$ is said to be convexif, for each $p, q \in E,$ it contains each shortest geodesic joining $p$ and $q.$A surface in $\mathbb{R}^3$ is said to have the fixed point property ifeach continuous mapping $T:E\rightarrow E$ from a compact convex subset $E$ of $X$ has a fixed point.In this paper, we give some examples of surfaces in $\mathbb{R}^3$ that do not have the fixed point property.Moreover, we show that the surface $z=y^2$ and the upper hemisphere of the sphere of radius $r$ centered at $(0, 0, 0)$ with north pole and equator removed have the fixed point property.