Morse index
A number associated with a critical point of a smooth function on a manifold or of a geodesic on a Riemannian (or Finsler) manifold.
1) The Morse index of a critical point of a smooth function
on a manifold
is equal, by definition, to the negative index of inertia of the Hessian of
at
(cf. Hessian of a function), that is, the dimension of the maximal subspace of the tangent space
of
at
on which the Hessian is negative definite. This definition makes sense also for twice (Fréchet) differentiable functions on infinite-dimensional Banach spaces. The only difference is that the value
is admissible for the index. In this case it is expedient to introduce the idea of the co-index of a critical point
of
as the positive index of inertia of the Hessian (the second Fréchet differential) of
at
.
2) Let and
be smooth submanifolds of a complete Riemannian space
. For a piecewise-smooth path
with
,
, transversal to
and
at its end-points
and
, the analogue of a tangent space is the vector space
of all piecewise-smooth vector fields
along
for which
,
. For any geodesic
with
, orthogonal at its end-points
and
to
and
, respectively, the second variation
of the action functional (see Morse theory) defines a symmetric bilinear functional
on
(the analogue of the Hessian). The Morse index of the geodesic is equal, by definition, to the negative index of inertia of this functional. The null space
of
on
(the set of
at which
for all
) consists exactly of the Jacobi fields (cf. Jacobi vector field)
. If
, the geodesic is called
-degenerate, and
is called the order of degeneracy of the geodesic.
The case when is a point
is considered below. Let
be the normal bundle to
in
and let
be its fibre over
. The restriction of the exponential mapping
defines a mapping
. A geodesic
,
,
, is
-degenerate if and only if the kernel of the differential
of
at
is not null; in this connection, the dimension of the kernel is equal to the order of degeneracy of the geodesic
. A point
,
, is called a focal point of
along
if the geodesic
is
-degenerate; the order of degeneracy of
is called the multiplicity of the focal point
. By the Sard theorem, the set of focal points has measure zero, so a typical geodesic is non-degenerate. If
also consists of one point
(
is not excluded), then a focal point is called adjoint to
along
. The Morse index theorem [1] asserts that the Morse index of a geodesic is finite and equal to the number of focal points
of
,
, taking account of multiplicity.
References
[1] | M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) |
[2] | W. Ambrose, "The index theorem in Riemannian geometry" Ann. of Math. , 73 (1961) pp. 49–86 |
Comments
There is a natural generalization of the Morse index of geodesics to variational calculus, which runs as follows. Let be a real-valued smooth function on an open subset
of
and let
be a smooth submanifold of
. Let
be the space of smooth curves
for which the
-jet lies in
and
. Then
is a Banach manifold, on which one has the smooth functional
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One then considers the Morse index of at critical curves
; it is finite if the Hessian of
is positive definite at
,
,
(Legendre's condition, cf. Legendre condition).
References
[a1] | J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) |
[a2] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) |
[a3] | W. Klingenberg, "Lectures on closed geodesics" , Springer (1978) |
Morse index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_index&oldid=47903