The generalized Morse lemma and the Euler characteristic on Banach manifolds
Let (E, \(\| \|)\) be a Banach space endowed with a scalar product \(<, >\) which induces on E a norm \(\| \|_ w\) weaker than \(\| \|\). If M is a Finsler manifold modeled on E, it is possible to consider some classes of functions f: \(M\to {\mathbb{R}}\) for which the main results of Morse theory are still valid. The authors give an extension in this setting of a result of \textit{D. Gromoll} and \textit{W. Meyer} [Topology 8, 361-369 (1969; Zbl 0212.289)]. If p is an isolated critical point of f, then near p the following decomposition holds: \[ f\circ \psi (x+y)=(1/2)d^ 2f_ p(x,x)+f^ 0(y), \] where \(\psi\) is a local diffeomorphism and \(f^ 0\) is a function defined on the kernel of \(d^ 2f_ p\). This result is then applied in order to show that the Euler characteristic of M is equal to the Euler characteristic of the vector field \(\nabla f\).
scientific article; zbMATH DE number 4111627