# Extension property of continuous functions in a Riemannain manifold with a pole

## TL;DR

This paper generalizes the Brouwer fixed point theorem to 2-dimensional Riemannian manifolds with a pole, showing that continuous functions on the boundary of convex domains can be extended inside without fixed points.

## Contribution

It introduces a new extension property for continuous functions in Riemannian manifolds with a pole, expanding fixed point results beyond Euclidean spaces.

## Key findings

- Fixed point existence on boundary of convex domains
- Extension of boundary functions inside convex domains
- Generalization of Brouwer's theorem to Riemannian manifolds

## Abstract

The Brouwer fixed point theorem says that any continuous function from disc to itself has a fixed point. By using simple geometrical technique we have generalized the result in manifold and proved that any continuous function on the boundary of a bounded convex domain of a $2$-dimensional Riemannian manifold with a pole having at least one fixed point can be extended to the convex domain without any interior fixed point.