Let's consider a simple closed geodesic on a surface beginning at some point P with initial unit velocity vector V. For any initial unit vector W close to V there will be a unique geodesic which returns to the vicinity of P. This describes a "first return" map which takes a transverse section (in the unit tangent bundle) to the geodesic to itself. The linearized version is the Poincare map, and the geodesic is non-degenerate if this linearized map does not have 1 as an eigenvalue. Since the map turns out to be area-preserving, the product of the eigenvalues is 1, so in the non-degenerate case the eigenvalues are either complex of absolute value 1 (the elliptic case) or real (the hyperbolic case). A nice example of a non-degenerate geodesic is the shortest closed geodesic on a compact surface of negative curvature; it will be hyperbolic. On a convex surface a typical short simple closed geodesic will be elliptic.

In other, word it seems that, $h=0$ is the only solution of

$$h''+ \sigma V^{-1}V_{\theta} h' + [(\lambda_1+ V_{\theta, \theta }) V^{-1}+ (\sigma^{-1}+1)k^2]h=0 $$ admits a solution where $h(0)= h(l)$ and $h'(0)= h'(l).$ Here V and k are periodic functions of \theta and V is a smooth bounded from below by a positive constant. \sigma>0.

Is it possible to show that $h=0$, if we only know that V>0.

A geodesic with fixed end points a and b is non-degenerate (i.e. has a non degenerate index form on the space of normal smooth vector fields), iff the endpoints are not conjugate. In this case the geodesic in question is the unique geodesic from a to b in a neighbourhood. In this setting 'most geodesics' are non-degenerate and, in particular, all 'short' geodesics are non-degenerate (short means shorter than the injectivity radius).

For a closed (i.e. perodic) geodesic the situation is slightly different; it is non-degenerate, if there are no periodic Jacobi field independent of the velocity field along the geodesic. On the sphere with the standard metric all closed geodesics are degenerate. A generic metric, however, would be "bumpy" such that all closed geodesics are non-degenerate (generic in the sense of Baire, a residual set of Riemannian metrics satisfies this property), as suggested by Abraham and proved by Anosov (1984).