Topological relations among the number of critical points of various orders (zero order are minima, first order are saddle points, etc.) are presented, which are invariants of the actual periodic (hyper)surfaces considered. The principal results involve the Betti numbers, the minimum number of critical points of a given order that must be present, and the Euler characteristic, based on an alternating sum rule involving the number of critical points of each order. Several conformational potential energy (hyper)surfaces are analyzed in terms of these relationships, including two containing lines of degeneracy and degenerate critical points.
ASJC Scopus subject areas
- Condensed Matter Physics
- Physical and Theoretical Chemistry