Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (Formula presented.) and (Formula presented.), represented by the Riemann surfaces which intersect along the curve (Formula presented.) were addressed. Curve (Formula presented.) does not contain geodesic lines in either of the manifolds (Formula presented.) and (Formula presented.). Consider six points located on the (Formula presented.)   (Formula presented.). The points (Formula presented.) are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (Formula presented.) /red links, and, alternatively, with the geodesic lines belonging to the manifold (Formula presented.) /green links. Points (Formula presented.) form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.