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A number associated with a [[Critical point|critical point]] of a smooth function on a manifold or of a geodesic on a Riemannian (or Finsler) manifold.
 
A number associated with a [[Critical point|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649601.png" /> of a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649602.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649603.png" /> is equal, by definition, to the negative index of inertia of the Hessian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649604.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649605.png" /> (cf. [[Hessian of a function|Hessian of a function]]), that is, the dimension of the maximal subspace of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649607.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649608.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m0649609.png" /> is admissible for the index. In this case it is expedient to introduce the idea of the co-index of a critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496011.png" /> as the positive index of inertia of the Hessian (the second Fréchet differential) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496013.png" />.
+
1) The Morse index of a critical point $  p $
 +
of a smooth function $  f $
 +
on a manifold $  M $
 +
is equal, by definition, to the negative index of inertia of the Hessian of $  f $
 +
at $  p $(
 +
cf. [[Hessian of a function|Hessian of a function]]), that is, the dimension of the maximal subspace of the tangent space $  T M _ {p} $
 +
of $  M $
 +
at $  p $
 +
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 $  + \infty $
 +
is admissible for the index. In this case it is expedient to introduce the idea of the co-index of a critical point $  p $
 +
of $  f $
 +
as the positive index of inertia of the Hessian (the second Fréchet differential) of $  f $
 +
at $  p $.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496015.png" /> be smooth submanifolds of a complete Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496016.png" />. For a piecewise-smooth path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496017.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496019.png" />, transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496021.png" /> at its end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496023.png" />, the analogue of a tangent space is the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496024.png" /> of all piecewise-smooth vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496025.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496028.png" />. For any geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496030.png" />, orthogonal at its end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496032.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496034.png" />, respectively, the second variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496035.png" /> of the action functional (see [[Morse theory|Morse theory]]) defines a symmetric bilinear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496037.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496040.png" /> (the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496041.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496043.png" />) consists exactly of the Jacobi fields (cf. [[Jacobi vector field|Jacobi vector field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496045.png" />, the geodesic is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496047.png" />-degenerate, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496048.png" /> is called the order of degeneracy of the geodesic.
+
2) Let $  V _ {0} $
 +
and $  V _ {1} $
 +
be smooth submanifolds of a complete Riemannian space $  M $.  
 +
For a piecewise-smooth path $  \omega :  [ 0 , 1 ] \rightarrow M $
 +
with $  \omega ( i) \in V _ {i} $,
 +
$  i = 0 , 1 $,  
 +
transversal to $  V _ {0} $
 +
and $  V _ {1} $
 +
at its end-points $  \omega ( 0) $
 +
and $  \omega ( 1) $,  
 +
the analogue of a tangent space is the vector space $  T _  \omega  = T _ {\omega , V _ {0}  , V _ {1} } $
 +
of all piecewise-smooth vector fields $  W $
 +
along $  \omega $
 +
for which $  W ( \omega ( i) ) \in ( T V _ {i} ) _ {\omega ( i) }  $,
 +
$  i = 0 , 1 $.  
 +
For any geodesic $  \gamma :  [ 0 , 1 ] \rightarrow M $
 +
with $  \gamma ( i) \in V _ {i} $,  
 +
orthogonal at its end-points $  \gamma ( 0) $
 +
and $  \gamma ( 1) $
 +
to $  V _ {0} $
 +
and $  V _ {1} $,  
 +
respectively, the second variation $  \delta  ^ {2} E $
 +
of the action functional (see [[Morse theory|Morse theory]]) defines a symmetric bilinear functional $  E _ {**} $
 +
on $  T _  \gamma  $(
 +
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 $  N _  \gamma  $
 +
of $  E _ {**} $
 +
on $  T _  \gamma  $(
 +
the set of $  X \in T _  \gamma  $
 +
at which $  E _ {**} ( X , Y ) = 0 $
 +
for all $  Y \in T _  \gamma  $)  
 +
consists exactly of the Jacobi fields (cf. [[Jacobi vector field|Jacobi vector field]]) $  J \in T _  \gamma  $.  
 +
If $  N _  \gamma  \neq 0 $,  
 +
the geodesic is called $  ( V _ {0} , V _ {1} ) $-
 +
degenerate, and $  \mathop{\rm dim}  N _  \gamma  $
 +
is called the order of degeneracy of the geodesic.
  
The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496049.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496050.png" /> is considered below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496051.png" /> be the [[Normal bundle|normal bundle]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496053.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496054.png" /> be its fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496055.png" />. The restriction of the [[Exponential mapping|exponential mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496056.png" /> defines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496057.png" />. A geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496060.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496061.png" />-degenerate if and only if the kernel of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496063.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496064.png" /> is not null; in this connection, the dimension of the kernel is equal to the order of degeneracy of the geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496065.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496067.png" />, is called a focal point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496068.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496069.png" /> if the geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496070.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496071.png" />-degenerate; the order of degeneracy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496072.png" /> is called the multiplicity of the focal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496073.png" />. By the [[Sard theorem|Sard theorem]], the set of focal points has measure zero, so a typical geodesic is non-degenerate. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496074.png" /> also consists of one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496075.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496076.png" /> is not excluded), then a focal point is called adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496077.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496078.png" />. The Morse index theorem [[#References|[1]]] asserts that the Morse index of a geodesic is finite and equal to the number of focal points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496081.png" />, taking account of multiplicity.
+
The case when $  V _ {1} $
 +
is a point $  q \in M $
 +
is considered below. Let $  \nu $
 +
be the [[Normal bundle|normal bundle]] to $  V = V _ {0} $
 +
in $  M $
 +
and let $  \nu ( p) $
 +
be its fibre over $  p \in V $.  
 +
The restriction of the [[Exponential mapping|exponential mapping]] $  T M \rightarrow M $
 +
defines a mapping $  \mathop{\rm exp} : \nu \rightarrow M $.  
 +
A geodesic $  \gamma ( t) = \mathop{\rm exp}  ( t \xi ) $,  
 +
$  \xi \in \nu ( p) $,  
 +
0 \leq  t \leq  1 $,  
 +
is $  ( V ,  \mathop{\rm exp}  \xi ) $-
 +
degenerate if and only if the kernel of the differential $  d _  \xi    \mathop{\rm exp} : T \nu _  \xi  \rightarrow T M _ { \mathop{\rm exp}  \xi } $
 +
of $  \mathop{\rm exp} $
 +
at $  \xi $
 +
is not null; in this connection, the dimension of the kernel is equal to the order of degeneracy of the geodesic $  \gamma $.  
 +
A point $  s = \gamma ( t _ {0} ) $,  
 +
$  0 < t _ {0} \leq  1 $,  
 +
is called a focal point of $  V $
 +
along $  \gamma $
 +
if the geodesic $  \gamma  ^  \prime  : t \rightarrow \gamma ( t / t _ {0} ) $
 +
is $  ( V , s ) $-
 +
degenerate; the order of degeneracy of $  \gamma $
 +
is called the multiplicity of the focal point $  s $.  
 +
By the [[Sard theorem|Sard theorem]], the set of focal points has measure zero, so a typical geodesic is non-degenerate. If $  V $
 +
also consists of one point $  p \in M $(
 +
$  p = q $
 +
is not excluded), then a focal point is called adjoint to $  p $
 +
along $  \gamma $.  
 +
The Morse index theorem [[#References|[1]]] asserts that the Morse index of a geodesic is finite and equal to the number of focal points $  \gamma ( t) $
 +
of  $  V $,
 +
0 < t < 1 $,  
 +
taking account of multiplicity.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,  "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Ambrose,  "The index theorem in Riemannian geometry"  ''Ann. of Math.'' , '''73'''  (1961)  pp. 49–86</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,  "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Ambrose,  "The index theorem in Riemannian geometry"  ''Ann. of Math.'' , '''73'''  (1961)  pp. 49–86</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
There is a natural generalization of the Morse index of geodesics to [[Variational calculus|variational calculus]], which runs as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496082.png" /> be a real-valued smooth function on an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496084.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496085.png" /> be a smooth submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496086.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496087.png" /> be the space of smooth curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496088.png" /> for which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496089.png" />-jet lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496091.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496092.png" /> is a Banach manifold, on which one has the smooth functional
+
There is a natural generalization of the Morse index of geodesics to [[Variational calculus|variational calculus]], which runs as follows. Let $  f $
 +
be a real-valued smooth function on an open subset $  Z $
 +
of $  [ 0 , 1 ] \times T M $
 +
and let $  R $
 +
be a smooth submanifold of $  M \times M $.  
 +
Let $  C $
 +
be the space of smooth curves $  \omega :  [ 0 , 1 ] \rightarrow M $
 +
for which the $  1 $-
 +
jet lies in $  Z $
 +
and $  ( \omega ( 0) , \omega ( 1) ) \in R $.  
 +
Then $  C $
 +
is a Banach manifold, on which one has the smooth functional
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496093.png" /></td> </tr></table>
+
$$
 +
F : \omega  \mapsto  \int\limits _ { 0 } ^ { 1 }
 +
f \left ( t , \omega ( t) ,
 +
\frac{d \omega }{dt}
 +
( t) \right )  d t .
 +
$$
  
One then considers the Morse index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496094.png" /> at critical curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496095.png" />; it is finite if the Hessian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496096.png" /> is positive definite at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064960/m06496099.png" /> (Legendre's condition, cf. [[Legendre condition|Legendre condition]]).
+
One then considers the Morse index of $  F $
 +
at critical curves $  \omega $;  
 +
it is finite if the Hessian of $  v \mapsto f ( t , x , v ) $
 +
is positive definite at $  x = \omega ( t) $,
 +
$  v = ( d \omega / dt) ( t) $,  
 +
$  t \in [ 0 , 1 ] $(
 +
Legendre's condition, cf. [[Legendre condition|Legendre condition]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Milnor,  "Morse theory" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Klingenberg,  "Lectures on closed geodesics" , Springer  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Milnor,  "Morse theory" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Klingenberg,  "Lectures on closed geodesics" , Springer  (1978)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


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 $ p $ of a smooth function $ f $ on a manifold $ M $ is equal, by definition, to the negative index of inertia of the Hessian of $ f $ at $ p $( cf. Hessian of a function), that is, the dimension of the maximal subspace of the tangent space $ T M _ {p} $ of $ M $ at $ p $ 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 $ + \infty $ is admissible for the index. In this case it is expedient to introduce the idea of the co-index of a critical point $ p $ of $ f $ as the positive index of inertia of the Hessian (the second Fréchet differential) of $ f $ at $ p $.

2) Let $ V _ {0} $ and $ V _ {1} $ be smooth submanifolds of a complete Riemannian space $ M $. For a piecewise-smooth path $ \omega : [ 0 , 1 ] \rightarrow M $ with $ \omega ( i) \in V _ {i} $, $ i = 0 , 1 $, transversal to $ V _ {0} $ and $ V _ {1} $ at its end-points $ \omega ( 0) $ and $ \omega ( 1) $, the analogue of a tangent space is the vector space $ T _ \omega = T _ {\omega , V _ {0} , V _ {1} } $ of all piecewise-smooth vector fields $ W $ along $ \omega $ for which $ W ( \omega ( i) ) \in ( T V _ {i} ) _ {\omega ( i) } $, $ i = 0 , 1 $. For any geodesic $ \gamma : [ 0 , 1 ] \rightarrow M $ with $ \gamma ( i) \in V _ {i} $, orthogonal at its end-points $ \gamma ( 0) $ and $ \gamma ( 1) $ to $ V _ {0} $ and $ V _ {1} $, respectively, the second variation $ \delta ^ {2} E $ of the action functional (see Morse theory) defines a symmetric bilinear functional $ E _ {**} $ on $ T _ \gamma $( 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 $ N _ \gamma $ of $ E _ {**} $ on $ T _ \gamma $( the set of $ X \in T _ \gamma $ at which $ E _ {**} ( X , Y ) = 0 $ for all $ Y \in T _ \gamma $) consists exactly of the Jacobi fields (cf. Jacobi vector field) $ J \in T _ \gamma $. If $ N _ \gamma \neq 0 $, the geodesic is called $ ( V _ {0} , V _ {1} ) $- degenerate, and $ \mathop{\rm dim} N _ \gamma $ is called the order of degeneracy of the geodesic.

The case when $ V _ {1} $ is a point $ q \in M $ is considered below. Let $ \nu $ be the normal bundle to $ V = V _ {0} $ in $ M $ and let $ \nu ( p) $ be its fibre over $ p \in V $. The restriction of the exponential mapping $ T M \rightarrow M $ defines a mapping $ \mathop{\rm exp} : \nu \rightarrow M $. A geodesic $ \gamma ( t) = \mathop{\rm exp} ( t \xi ) $, $ \xi \in \nu ( p) $, $ 0 \leq t \leq 1 $, is $ ( V , \mathop{\rm exp} \xi ) $- degenerate if and only if the kernel of the differential $ d _ \xi \mathop{\rm exp} : T \nu _ \xi \rightarrow T M _ { \mathop{\rm exp} \xi } $ of $ \mathop{\rm exp} $ at $ \xi $ is not null; in this connection, the dimension of the kernel is equal to the order of degeneracy of the geodesic $ \gamma $. A point $ s = \gamma ( t _ {0} ) $, $ 0 < t _ {0} \leq 1 $, is called a focal point of $ V $ along $ \gamma $ if the geodesic $ \gamma ^ \prime : t \rightarrow \gamma ( t / t _ {0} ) $ is $ ( V , s ) $- degenerate; the order of degeneracy of $ \gamma $ is called the multiplicity of the focal point $ s $. By the Sard theorem, the set of focal points has measure zero, so a typical geodesic is non-degenerate. If $ V $ also consists of one point $ p \in M $( $ p = q $ is not excluded), then a focal point is called adjoint to $ p $ along $ \gamma $. The Morse index theorem [1] asserts that the Morse index of a geodesic is finite and equal to the number of focal points $ \gamma ( t) $ of $ V $, $ 0 < t < 1 $, 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 $ f $ be a real-valued smooth function on an open subset $ Z $ of $ [ 0 , 1 ] \times T M $ and let $ R $ be a smooth submanifold of $ M \times M $. Let $ C $ be the space of smooth curves $ \omega : [ 0 , 1 ] \rightarrow M $ for which the $ 1 $- jet lies in $ Z $ and $ ( \omega ( 0) , \omega ( 1) ) \in R $. Then $ C $ is a Banach manifold, on which one has the smooth functional

$$ F : \omega \mapsto \int\limits _ { 0 } ^ { 1 } f \left ( t , \omega ( t) , \frac{d \omega }{dt} ( t) \right ) d t . $$

One then considers the Morse index of $ F $ at critical curves $ \omega $; it is finite if the Hessian of $ v \mapsto f ( t , x , v ) $ is positive definite at $ x = \omega ( t) $, $ v = ( d \omega / dt) ( t) $, $ t \in [ 0 , 1 ] $( 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)
How to Cite This Entry:
Morse index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_index&oldid=47903
This article was adapted from an original article by M.M. PostnikovYu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article