On the creation of conjugate points (Q1183067)

For a differentiable manifold \(M\), let \(B(M)\) be the space of Riemannian metrics on \(M\) with no conjugate points, and \(A(M)\) the space of metrics on \(M\) whose geodesic flows are Anosov. \(A(M)\) is open in the \(C^ k\) topology for every \(k\in \mathbb{N}\) (Anosov), and \(A(M)\subset B(M)\) by Klingenberg. Thus, if \(M\) admits a metric with Anosov geodesic flow then the interior of \(B(M)\) in the \(C^ k\) topology (for every \(k\in \mathbb{N}\)) is non-empty. The author shows: \(A(M)\) is the interior of \(B(M)\) in the \(C^ 2\) topology. The proof is given by the following existence theorem: for \((M,g)\) a Riemannian manifold with no conjugate points whose geodesic flow is not Anosov, given any \(\varepsilon > 0\), there exists a metric \(g_ \varepsilon\), conformal to \(g\), having conjugate points, and \(\| g-g_ \varepsilon\|_{C^ 2} < \varepsilon\).