Isoperimetric problems for rotation functionals of the first and second orders in (pseudo) Riemannian manifolds (Q881222)

For a pseudo-Riemannian manifold \((M,g)\) and a curve \(\gamma : t\rightarrow \gamma(t)\in M\), let \(\xi=\dot\gamma\), \(\xi_ 1=\nabla\xi\), and \(\xi_ 2=\nabla\xi_ 1\). Let \(L_ 1(x,\xi,\xi_ 1)\) and \(L_ 2(x,\xi,\xi_ 1, \xi_ 2)\) be the Lagrangians of rotational functionals of the first and second order, respectively. In this paper, the author studies variational problems for degenerate Lagrangians on pseudo-Riemannian manifolds and proves the singularity of \(L_ 1\) and \(L_ 2\). Furthermore, various isoperimetric problems related to rotational functionals are considered and some results for such functionals on surfaces are obtained.