## Trajectories of dynamical systems joining two given submanifolds.(English)Zbl 0853.58033

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.

### MSC:

 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces