On a compact Riemannian manifold, A/, there ought to be infinitely many
closed geodesies (a classical conjecture). This is obvious if the isometry group of
M has dimension greater than zero, so we should examine the "generic case" of
minimal symmetry. For example, suppose M is a 2-sphere embedded in 3-space,
with the induced metric. In the case of the standard embedding, every point is in
a 1-parameter family of closed geodesies. But if the embedding is perturbed to an
ellipsoid with unequal axes, most of these geodesies disappear. Three short ones
remain, and there are arbitrarily long spiraling ones as well. Perturbed further by
bumps or undulations, this is an example of a bumpy metric, with a countable set
of closed orbits (finitely many of bounded length), all stable in some sense, and a
O-dimensional isometry group. The definition is stated later, and the main theorem:
on a compact manifold, almost all metrics are bumpy. We conjecture that every
bumpy metric on a compact manifold has infinitely many distinct closed geodesies,
and this has been proved for some manifolds by Gromoll and W. Meyer [4].
1. The definition of bumpy. We consider closed geodesies from the point of view
of Marston Morse. Let H = Hl(S\M) be the Hilbert manifold of absolutely
continuous maps c:S! -• M, and Jg :H -+ R the energy function for a Riemannian
metricg on M (see Palais [6]). Then a critical point ceH of Jg is a closed geodesic
parameterized proportionately to arclength (we identify
% [0, l]/{0,1}). As
the group
acts continuously on H by 9(c)(t) = c(6 + t) for (0, c, t) e
x H x
and Jg is invariant under this action, the orbit S^c) of a critical point ceH consists
entirely of critical points. In fact if c e H is a
curve, we may prove that
is a
submanifold of H, so this always occurs for critical points (by the usual
regularity theorem). Then S*(c) is a critical manifold of Jg corresponding to a
single closed geodesic in M.
A Riemannian metric g on a manifold M is bumpy iff for every
nonconstant critical point ceH of Jg, Sl(c) is a nondegenerate critical manifold
of Jg. That is, the index form (or Hessian
has a 1-dimensional null space,
the tangent space of
at c.
The constant curves of H are excluded because they comprise a submanifold of
H diffeomorphic to M, which is a nondegenerate critical manifold of Jg (at least if
M is compact) of higher dimension. Because of Bott's formula relating the nullity
of an iterated closed geodesic to its Poincare rotation numbers [3], this definition
is equivalent to: every closed geodesic has irrational rotation numbers.
2. Properties of bumpy metrics. Ifg is a C+4 metric on M, then dJg is a C section
of the cotangent bundle T*H. Let H0 c H denote the closure of the constant
Previous Page Next Page