On the existence of infinitely many periodic solutions of non-complete Riemannian manifolds
Let \(M\) be a Riemannian manifold and \(V : M \times \mathbb{R} \to \mathbb{R}\) a \(C^1\) potential function. The author considers curves \(x : [0,T] \to {\mathcal M}\) satisfying the equation \[ D_t \dot x(t) = -\nabla_\mathbb{R} V(x (t), t), \tag{*} \] where \(D_t \dot x(t)\) denotes the covariant derivative of \(\dot x(t)\) along the direction \(\dot x(t)\) and \(\nabla_\mathbb{R} V(x(t), t)\) the Riemannian gradient with respect to \(x\) of \(V\) at \((x(t), t)\). In the present paper the author studies two problems: (a) for periodic \(V\) find the periodic solutions of (*); (b) for given \(x_1, x_2 \in {\mathcal M}\) find the solutions of (*) when \(x(0) = x_1\) and \(x(T) = x_2\).
scientific article; zbMATH DE number 790888