Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distrib\-utions (Q1411260)

Within the framework of variational theory for curves that are minimizers of sub-Riemannian length functionals, the authors develop the basis of a Morse theory for sub-Riemannian normal geodesics, i.e., for curves that are stationary for the sub-Riemannian energy functional. More specific, considering a Riemannian manifold \((M,g)\), a unit vector field \(Y\) and a codimension one distribution \(\Delta\subset TM\) on \(M\) which is orthogonal to \(Y\), the horizontal curves (i.e., the curves everywhere tangent to \(\Delta\)) which join a smooth submanifold \(P\) of \(M\) and a fixed point \(q\in M\) are investigated. The second variation of the energy \(E\) is computed at its critical points. The notions of \(P\)-Jacobi field, \(P\)-focal point and of exponential map are introduced, and a Morse index theorem is proved. As well, the Morse relations for \(E\) are derived, under the assumption of completeness for \((M,g)\).