Examples of Compact Hypersurfaces in R^2P, 2P ≥ 6, With No Periodic Orbits

Abstract

Let (M2p, w ₀) be a C∞ symplectic, metrisable, manifold. For example T*Rp, with its canonical symplectic form. Let H0: M→R be a C∞ function, c a regular value of H0. Then $H_0^{ - 1}$(c) is a codimension 1 C∞ manifold. Given any codimension 1 C∞ manifold N0 M2p, then N0 is a connected component of $H_0^{ - 1}$(c), for some C∞ function H0 and c as regular value.

Let $X_{H_0 }$ the Hamiltonian vectorfield associated to H0 and w0 such that $i_{X_{H_0 } } \,w_0 \, = \,dH_0$, and $f_t^{H_0 }$ the local Hamiltonian flow of $X_{H_0 }$, that leaves invariant Ho and N0. If we consider the various possible choices of H0, for a given N0, then changes toC∞ (N0,R), ≅ 0 (i.e., the paramctrisation of the orbits). Rcparametrisation does not change the periodic orbits but usually does change the periods.