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 $ 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)