Connectivities R(i) of Fréchet Spaces in Variational Topology

A global study of geodesics, joining points A(1) [unk] A(2) on a Riemannian manifold N(n), is oriented by the answers given to two questions. How does the concept of a nondegenerate geodesic enter, and what topological invariants best condition the geodesics joining A(1) to A(2)? We answer the first question by defining and exploiting nondegenerate point pairs A(1) [unk] A(2) on N(n). The connectivities R(i) of M(n) used in ordinary critical point theory should be replaced by the connectivities R(i) of the pathwise components of the Fréchet metric space [unk](A(1) ) (A(2) ) defined in this paper. These components are homeomorphic. Their connectivities R(i) condition the geodesics joining A(1) to A(2) on N(n) in a way to be disclosed. Detailed proofs are found in the references listed, or will be presented later.