  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.

Differential Geometry