Morse theory for Riemannian geodesics without nondegeneracy assumptions
type: Item
instance of: scholarly article
title: Morse theory for Riemannian geodesics without nondegeneracy assumptions
zbMATH Open document ID: 0922.58010
DOI: 10.12775/TMNA.1997.034
published in: Topological Methods in Nonlinear Analysis
publication date: 1999-10-10T00:00:00Z
review text: Let \(M\) be a Hilbert manifold and \(f\in C^2(M,\mathbb R)\) be bounded from below, satisfying the Palais-Smale condition and such that the Hessian of \(f\) at a critical point is Fredholm of index \(0\). The critical points need not be isolated. In this situation the authors define a generalized Morse index of \(f\). The definition is based on considering the set of classical Morse polynomials of certain perturbations of \(f\) which are Morse functions. The classical Morse inequalities generalize immediately. The abstract theory is applied to the theory of geodesics joining two points on a finite-dimensional Riemannian manifold. Here \(f\) is the Dirichlet integral. The authors also introduce a geometric Morse index of \(f\) based on the notion of conjugate points for critical points of perturbations of \(f\).
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 1192343
zbMATH Keywords: abstract critical point theory
zbMATH Keywords: generalized Morse relations
zbMATH Keywords: perturbation of functionals
zbMATH Keywords: theory of geodesics
Wikidata QID: Q115239579
author: Anna Valeria Germinario
author: Fabio Giannoni
reviewed by: Thomas J. Bartsch
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1192343