  Deciding whether saddle points exist or are approximable for
nonconvex-nonconcave problems is usually intractable. This paper takes a step
towards understanding a broad class of nonconvex-nonconcave minimax problems
that do remain tractable. Specifically, it studies minimax problems over
geodesic metric spaces, which provide a vast generalization of the usual
convex-concave saddle point problems. The first main result of the paper is a
geodesic metric space version of Sion's minimax theorem; we believe our proof
is novel and broadly accessible as it relies on the finite intersection
property alone. The second main result is a specialization to geodesically
complete Riemannian manifolds: here, we devise and analyze the complexity of
first-order methods for smooth minimax problems.
