Geometric-Topological Studies on the Generalization of Gradient Flows and Morse Relations for Non-Closed Manifolds

Abstract

In this paper, we give a rigorous geometric and topological framework that generalizes classical Morse theory, gradient flows and Poincaré–Hopf index formulas to non-closed manifolds, in particular compact manifolds with boundary (). The Euler characteristic  of a compact, closed and oriented manifold is the alternating sum of its Morse indices and Betti numbers as established by classical Morse theory. For manifolds with boundary, however, gradient trajectories may reach the boundary or cross it, and thus standard flow-diffeomorphism proofs and classical cell attachments may be invalid. In this revised manuscript we address all evaluation comments of reviewers: (1) Corrected Homological Calculations for Relative Topological Invariants:  in the case of the smooth cylinder  and the Möbius strip; (2) Boundary-Transversality Conditions and Local Quadratic Normal Forms: classified boundary critical points as inward-pointing and outward-pointing; (3) Formulation and proof of an original main theorem—the Boundary-Relative Morse-Euler Equivalence Theorem (Theorem 4.10) which rigorously establishes the identity:

(4) Complete and corrected geometric calculations for cylinders, Möbius strips, and 2-disks, using explicit outward normal vectors and non-degenerate functions; and (5) Detailed Comparative Discussion that connects our relative setting with classical Morse and relative index theory.