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)