Trajectories of dynamical systems joining two given submanifolds
Let \(({\mathcal M}, \langle \cdot,\cdot\rangle_R)\) be a Riemannian manifold, \(M_0\) and \(M_1\) closed submanifolds of \(\mathcal M\) and \(V : {\mathcal M} \to \mathbb{R}\) a \(C^1\) potential function. The author considers curves \(x : [0,1] \to {\mathcal M}\) orthogonal to \(M_0\) and \(M_1\) and satisfying \[ D_t \dot x(t) = -\nabla_R V(x(t)) \] with the boundary conditions \(x(0) \in M_0\), \(x(1) \in M_1\), \(\dot x (0) \in T_{x(0)} (M_0)\) and \(\dot x(1) \in T_{x(1)} (M_1)\). Here \(D_t \dot x(t)\) denotes the covariant derivative of \(\dot x (t)\) along the direction of \(x(t)\) and \(\nabla_R V(x(t))\) is the Riemannian gradient with respect to \(x\) of \(V\) in \(x(t)\). Using Lyusternik-Schnirelman theory, the author proves the existence of infinitely many such curves under a variety of reasonable conditions.
scientific article; zbMATH DE number 883505
scientific article