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

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.

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