The Bisection Theorem says that, given two regions in the plane, there exists a line separating them into equal size respectively. This result has a generalization in $n$-dimensional Euclidean space, which can be proved by Borsuk Ulam Theorem.
I want to know whether this can be generalized to surfaces of not constant curvature zero. For example, given two regions in hyperbolic paraboloid surface, does there exist a geodesic line that separates them into equal size respectively?