Conjugate points are critical values of the exponential map along the geodesic

Define $\mathrm{\Phi }:T_{p}M\to \mathcal{J}(\gamma )$ by $\mathrm{\Phi }(w)=J_w$ for the unique Jacobi field of [F3] with $J_w(0)=0$ and $D_t{J}_w(0)=w$. Then $\mathrm{\Phi }$ is real-linear: for $a,b\in \mathbb{R}$ and $u,w\in T_{p}M$, the field $a{J}_u+b{J}_w$ is Jacobi by [F2], and its initial data at $0$ are $0$ and $au+bw$ by [F2], so uniqueness in [F3] gives $a{J}_u+b{J}_w=J_{au+bw}$, that is, $\mathrm{\Phi }(au+bw)=a\mathrm{\Phi }(u)+b\mathrm{\Phi }(w)$.

The map $\mathrm{\Phi }$ is injective: if $\mathrm{\Phi }(w)=0$, then $w=D_t{J}_w(0)=D_{t}0(0)=0$. Moreover $\mathrm{\Phi }(T_{p}M)=\{J\in \mathcal{J}(\gamma ):J(0)=0\}$: for such a $J$, put $w=D_{t}J(0)$; then $J$ and $J_w$ are Jacobi fields with the same position and derivative at $0$, so $J=J_w=\mathrm{\Phi }(w)$ by [F3].

By [F4], for every $w\in T_{p}M$, $d({\exp }_p{)}_v(w)=\mathrm{\Phi }(w)(1)\mathrm{.}$ Hence, using [F1], $w\in \ker d({\exp }_p{)}_v⟺\mathrm{\Phi }(w)\in {\mathcal{K}}_{\gamma }(0,1)\mathrm{.}$ Since $\mathrm{\Phi }$ is injective and ${\mathcal{K}}_{\gamma }(0,1)\subseteq \{J:J(0)=0\}=\mathrm{\Phi }(T_{p}M)$ by step 2.1, the restriction $\mathrm{\Phi }{\mathrm{\mid }}_{\ker d({\exp }_p{)}_v}:\ker d({\exp }_p{)}_v⟶{\mathcal{K}}_{\gamma }(0,1)$ is a well-defined bijection: it maps a $w$ with $d({\exp }_p{)}_v(w)=0$ to the field $\mathrm{\Phi }(w)$, which lies in ${\mathcal{K}}_{\gamma }(0,1)$ by the displayed equivalence, and it is surjective because every $J\in {\mathcal{K}}_{\gamma }(0,1)$ equals $\mathrm{\Phi }(w)$ with $w=D_{t}J(0)$ by step 2.1. It is linear as a restriction of a linear map.

The kernel of $d({\exp }_p{)}_v$ is a linear subspace of $T_{p}M$ by [F6], so by step 3.1 the multiplicity ${\dim }_{\mathbb{R}}{\mathcal{K}}_{\gamma }(0,1)$ equals $\dim \ker d({\exp }_p{)}_v$; in particular both spaces are finite-dimensional. If ${\mathcal{K}}_{\gamma }(0,1)\mathrm{\ne }\{0\}$ then, by [F1], $\gamma (0)$ and $\gamma (1)$ are conjugate and step 3.1 gives a nonzero element of $\ker d({\exp }_p{)}_v$, so the differential is not injective. Conversely, if the differential is not injective, step 3.1 and injectivity of $\mathrm{\Phi }$ give a nonzero element of ${\mathcal{K}}_{\gamma }(0,1)$, and [F1] makes the endpoints conjugate. Since the source and target of the differential both have dimension $n$ by [F5], [F6] identifies noninjectivity with singularity. This proves claims 1 and 2, the second in the form "the multiplicity equals the kernel dimension".

It remains to translate singularity into failure of local invertibility. Suppose first that $d({\exp }_p{)}_v$ is invertible. Choose a linear isomorphism of $T_{p}M$ onto ${\mathbb{R}}^n$ and a chart of $M$ around $\gamma (1)$; in these coordinates ${\exp }_p$ is a smooth map from a neighbourhood of $v$ in ${\mathbb{R}}^n$ to ${\mathbb{R}}^n$ whose derivative at $v$ is invertible, so [F8] makes ${\exp }_p$ a diffeomorphism from a neighbourhood of $v$ onto an open set. Conversely, suppose ${\exp }_p$ is a local diffeomorphism at $v$: there are open neighbourhoods $V$ of $v$ and $W$ of ${\exp }_p(v)=\gamma (1)$ such that ${\exp }_p{\mathrm{\mid }}_V:V\to W$ is a diffeomorphism. Its inverse $g:W\to V$ is smooth, and the chain rule [F8] applied to $g\circ ({\exp }_p{\mathrm{\mid }}_V)={\mathrm{i}\mathrm{d}}_V$ at $v$ gives $dg(\gamma (1))\circ d({\exp }_p{)}_v={\mathrm{i}\mathrm{d}}_{T_{p}M}$, so $d({\exp }_p{)}_v$ has a left inverse, hence is injective, hence bijective by [F6] and [F7]. Thus ${\exp }_p$ fails to be a local diffeomorphism at $v$ exactly when $d({\exp }_p{)}_v$ is singular, which by step 4.1 happens exactly when the endpoints are conjugate.

Boundary and choice audit. Since $v\mathrm{\ne }0$, the geodesic $\gamma$ has initial velocity $d({\exp }_p{)}_0(v)=v\mathrm{\ne }0$ and is nonconstant; the constant-geodesic clause of [F1] is therefore not needed. If $\dim M=0$ then $T_{p}M=\{0\}$ by [F5] and there is no nonzero $v$, so the theorem is vacuous; in dimension one every nonzero $v$ is in the single line $T_{p}M$ and the argument uses no splitting into normal and tangential directions. The zero vector $w=0$ has $\mathrm{\Phi }(0)=0$ and $d({\exp }_p{)}_v(0)=0$, consistent with [F4]; the case $w\mathrm{\ne }0$ is where the kernel is tested. Both directions of the equivalence in claim 1 and both directions of claim 3 are proved: conjugacy is derived from a nonzero kernel element in step 4.1, and singularity is derived from conjugacy there as well. The only choice assumption is [A1]; no selection from a family is made. [A1, F1, F4, F5, F9, step 4.1, step 5.1] $\mathrm{\square }$