Conditional extremals in complete Riemannian manifolds (Q412682)

In this paper the authors study conditional extremal curves in a complete Riemannian manifold \(M\) with prior vector field \(A\) on \(M\), i.e. critical points of the functional \(S(x)=\int_0^1 \langle \dot{x}-A, \dot{x}-A\rangle dt\) on the space of \(H^1\) curves in \(M\).NEWLINENEWLINELet \(\Omega_N(M)=\{ x\in H^1([0,1],M)/\; x(0)\in N_0, \;x(1)\in N_1\}\). If \(N_0\) and \(N_1\) are compact submanifolds of \(M\) and \(\|A\|_0\) is bounded, the authors prove that \(S\) satisfies the Palais-Smale condition on \(\Omega_N(M)\). Then by the Ljusternik-Schnirelman theory, the authors prove a Morse index theorem in the case \(N_0\) and \(N_1\) are single points, and use the Morse inequalities to obtain lower bounds on the number of critical points of each index.