Topological obstructions to integration by redundant set of integrals (Q1277591)

Let \( M \) be a compact \( n -\)dimensional manifold and \( g _{ij}\) a Riemannian metric on it. The Riemannian structure determines a canonical Hamiltonian system on \( T ^\star M \) via the function \(H=\frac{1}{2} \Sigma g _{ij} p ^i p ^j\), usually called geodesic flow. In the paper under review, the author proves the following result: Let us suppose that the geodesic flow has \( n+k-1 \) independent integrals so that \( B: \approx T ^{n-k} \times D ^{n+k-1}. \) Then, the first Betti number of \( M \) satisfies the following inequality: \(b _1 (M) \leq n-k\).