The Morse index theorem for geodesics

This pair proves the two finite-variation facts on which the local theory of conjugate points is built: the index lemma (Index lemma) and the loss of minimality past the first conjugate point (A geodesic does not minimize past its first conjugate point). The classical Morse index theorem is the quantitative refinement that turns those two facts into an exact count. It is recorded here as orientation only: it is not proved anywhere in this pair, no item of this pair or of the companion examples page uses it as a supplier, and no proof below quotes it as a proved statement.

In the formulation of the source, for a geodesic segment $\gamma :[a,b]\to M$ of the Levi-Civita connection, "the index of a geodesic segment is defined to be the maximum dimension of a linear space of proper normal vector fields on which $I$ is negative definite", and "the index of any geodesic segment is finite, and is equal to the number of its interior conjugate points counted with multiplicity" (Lee, printed p.189; here proper means that the fields vanish at $a$ and $b$).

In the notation of this pair the form is the index form of Index form of a geodesic segment restricted to the fixed-endpoint subspace ${\mathcal{X}}_0(\gamma )$ of continuous piecewise $C^1$ fields with $V(a)=V(b)=0$, and "counted with multiplicity" is the multiplicity of Conjugate points along a geodesic and their multiplicity. The count is over interior instants $t\in (a,b)$: the terminal instant $b$ is not counted, and a segment whose endpoint $\gamma (b)$ is conjugate to $\gamma (a)$ additionally carries positive nullity. The remark makes no claim of its own about the finiteness of the index or about either count; both are quoted from the source as orientation.

What this pair supplies in the direction of the theorem is exactly the two items named above. In the absence of conjugate instants, the index lemma produces the Jacobi field with prescribed endpoint values $J(a)=u$, $J(b)=w$, unique among Jacobi fields and satisfying $I_{\gamma }(J,J)\le I_{\gamma }(V,V)$ for every continuous piecewise $C^1$ competitor with the same endpoint values; and a conjugate instant $\gamma (c)$ with $a<c<b$ produces a piecewise smooth fixed-endpoint competitor on $[a,b]$ of strictly smaller energy and strictly smaller length. The full path-space statement of the Morse index theorem, together with the companion nullity statement, is deferred to a sequel.