The space of Jacobi fields along a geodesic has dimension two n

The smooth fields along $\gamma$ are closed under pointwise addition and real scalar multiplication: in a pulled-back frame these operations add and scale smooth coefficient functions by [F1]. The zero section and pointwise additive inverses are smooth too. The vector-space axioms hold in each tangent fiber by [F5] and [F6], hence hold pointwise for the space of smooth fields.

Write $V=T_{\gamma (a)}M$. By [F6] and [F7], choose a basis $e_1,\dots ,e_n$ of $V$, with the empty basis when $n=0$. Equip $V\times V$ with componentwise operations; every vector-space axiom follows in each component from [F5] and [F6]. Put $q_i=(e_i,0)$ and $r_i=(0,e_i)$ for $1\le i\le n$. These vectors span $V\times V$: expand each component in the basis of $V$. They are linearly independent because a vanishing linear combination has first and second components ${\sum }_i{a}_i{e}_i=0$ and ${\sum }_i{b}_i{e}_i=0$, forcing all $a_i,b_i$ to vanish by basis independence. Thus these $2n$ vectors form a basis of $V\times V$, also when $n=0$.

Put $T=\dot{\gamma }$ and define $L(J)=D_t^{2}J+R(J,T)T$. By [F3], its first term is real-linear in $J$; by [F4], its second term is pointwise real-linear in $J$. Thus $L(\lambda J+\mu K)=\lambda L(J)+\mu L(K)$ for smooth fields $J,K$ and real scalars $\lambda ,\mu$. By [F2], $\mathcal{J}(\gamma )=\{J:L(J)=0\}$; it contains zero and is closed under addition and scalar multiplication. The inherited vector-space axioms from step 1.1 therefore make $\mathcal{J}(\gamma )$ a real vector space.

Define the initial-data map $E:\mathcal{J}(\gamma )\to T_{\gamma (a)}M\times T_{\gamma (a)}M$ by $E(J)=(J(a),D_{t}J(a))$. The target has the componentwise vector-space operations established in step 1.2. Endpoint evaluation and covariant differentiation make $E$ preserve every linear combination by [F3], so it is linear by [F8]. Fact [F9] says exactly that every pair in its target is attained by one Jacobi field, and that no two Jacobi fields have the same image. Hence $E$ is bijective.

By [F9], let $J_i$ and $K_i$ be the unique Jacobi fields with initial data $(e_i,0)$ and $(0,e_i)$, respectively. Their images under $E$ are $q_i$ and $r_i$. Since $E$ is a linear bijection by step 3.1, the fields $J_1,\dots ,J_n,K_1,\dots ,K_n$ are linearly independent and span $\mathcal{J}(\gamma )$: apply $E$ to a relation or to the difference between a field and the corresponding linear combination, then use that the $q_i,r_i$ are a basis. Therefore $\mathcal{J}(\gamma )$ has dimension $2n$. If $n=0$, [F9] gives the unique zero field and this basis is empty.

The proof applies to $n=1$ with its two-element product basis, and to constant $\gamma$ because [F2] and [F9] include that case; the zero field and zero initial data are included. The hypothesis that $I$ is nondegenerate is needed for $D_t$; if $a$ is an included endpoint, [F3] and [F9] use the stated one-sided derivatives. A supplied geodesic rules out the empty-manifold case. The construction uses one finite basis of a single tangent space and the uniquely determined fields from [F9], so it uses no choice axiom. The statement is a dimension assertion, not an iff claim.