[proofplan] We prove uniqueness first and then existence. For uniqueness: in any chart $(U, \varphi)$ with coordinates $(x_1, \dots, x_n)$ and Christoffel symbols $\Gamma^i_{jk}$, the Leibniz rule (i) together with consistency (ii) on the coordinate vector fields $\partial_{x_j}$ — which are bona fide vector fields on $U$ — forces an explicit coordinate formula for $\nabla_{\partial_t} Y$ on $\gamma^{-1}(U)$. Any two operators satisfying (i) and (ii) must therefore agree locally, hence globally. For existence: we take this forced coordinate formula as a local definition, verify it is independent of the choice of chart (the transition rule for Christoffel symbols absorbs the Jacobian terms exactly so as to make the formula covariant), and then verify properties (i) and (ii) hold by direct computation. [/proofplan]

[step:Reduce the problem to coordinate charts and write a generic section in a local frame]Let $t_0 \in (a, b)$, and choose a chart $(U, \varphi)$ on $M$ with $\gamma(t_0) \in U$, coordinates $(x_1, \dots, x_n)$, and coordinate vector fields $\partial_{x_1}, \dots, \partial_{x_n} \in \Gamma(TU)$. By continuity of $\gamma$, there exists an open interval $I \subset (a, b)$ with $t_0 \in I$ and $\gamma(I) \subset U$. Write the curve in coordinates as $\gamma(t) = (x_1(t), \dots, x_n(t))$ for $t \in I$, where each $x_k \in C^\infty(I)$. Any section $Y \in \Gamma(\gamma^* TM)$ restricts to a smooth map $Y\big|_I: I \to TM$ with $Y(t) \in T_{\gamma(t)}M$, and decomposes uniquely as \begin{align*} Y(t) = \sum_{j=1}^n Y_j(t)\, \partial_{x_j}\big|_{\gamma(t)}, \qquad t \in I, \end{align*} where each $Y_j \in C^\infty(I)$. The Christoffel symbols of $\nabla^{\mathrm{aff}}$ in this chart are the smooth functions $\Gamma^i_{jk} \in C^\infty(U)$ defined by \begin{align*} \nabla^{\mathrm{aff}}_{\partial_{x_k}} \partial_{x_j} = \sum_{i=1}^n \Gamma^i_{jk}\, \partial_{x_i}. \end{align*}[/step]

[guided]The question: if we want to uniquely pin down an operator $\nabla_{\partial_t}$ using (i) and (ii), why work in charts? Because (i) is a pointwise/fibrewise property expressed in the curve parameter $t$, and (ii) refers to global vector fields on $M$ — but within a chart $(U, \varphi)$, the coordinate vector fields $\partial_{x_j}$ are global vector fields on $U$ (after possibly shrinking to ensure they are defined). So in a chart we have access to both properties at once, and we can expand the section $Y$ in the coordinate basis to exploit Leibniz on the coefficients $Y_j$ and consistency on the basis $\partial_{x_j}$. One subtlety: coordinate vector fields $\partial_{x_j}$ live on $U$, not on all of $M$. We only need them to be smooth vector fields on a neighbourhood of $\gamma(t_0)$; this is exactly what chart-level vector fields provide. Property (ii) is formulated with "$Y$ agrees near $t_0$ with the restriction of a global vector field $X$", and a chart-defined vector field behaves the same way locally — we can extend it globally by multiplying by a bump function supported in $U$ if we wish, and that extension agrees with $\partial_{x_j}$ on a neighbourhood of $\gamma(t_0)$. So (ii) applies directly to the coordinate basis. Once $Y$ is written as $Y(t) = \sum_j Y_j(t)\,\partial_{x_j}|_{\gamma(t)}$, the next step combines Leibniz on the scalar coefficients $Y_j(t)$ with consistency on the basis fields $\partial_{x_j}$.[/guided]

[step:Force the coordinate formula from the Leibniz rule and consistency]Apply (i) to each summand $Y_j(t)\, \partial_{x_j}\big|_{\gamma(t)}$, where we view $\partial_{x_j}\big|_{\gamma(t)}$ as the section of $\gamma^* TM$ induced by the vector field $\partial_{x_j}$ on $U$: \begin{align*} \nabla_{\partial_t} Y\big|_t = \sum_{j=1}^n \left( Y_j'(t)\, \partial_{x_j}\big|_{\gamma(t)} + Y_j(t)\, \nabla_{\partial_t} \bigl(\partial_{x_j} \circ \gamma\bigr)\big|_t \right). \end{align*} The vector field $\partial_{x_j}$ is a smooth vector field on $U$, so property (ii) applies with $X = \partial_{x_j}$: \begin{align*} \nabla_{\partial_t}\bigl(\partial_{x_j} \circ \gamma\bigr)\big|_t = \nabla^{\mathrm{aff}}_{\gamma'(t)} \partial_{x_j}\big|_{\gamma(t)}. \end{align*} Expanding $\gamma'(t) = \sum_k x_k'(t)\, \partial_{x_k}\big|_{\gamma(t)}$ and using $C^\infty(U)$-linearity of $\nabla^{\mathrm{aff}}$ in its first argument together with the definition of Christoffel symbols, \begin{align*} \nabla^{\mathrm{aff}}_{\gamma'(t)} \partial_{x_j}\big|_{\gamma(t)} = \sum_{k=1}^n x_k'(t)\, \nabla^{\mathrm{aff}}_{\partial_{x_k}} \partial_{x_j}\big|_{\gamma(t)} = \sum_{i,k=1}^n \Gamma^i_{jk}(\gamma(t))\, x_k'(t)\, \partial_{x_i}\big|_{\gamma(t)}. \end{align*} Substituting back and relabelling dummy indices, \begin{align*} \nabla_{\partial_t} Y\big|_t = \sum_{i=1}^n \left( Y_i'(t) + \sum_{j,k=1}^n \Gamma^i_{jk}(\gamma(t))\, x_k'(t)\, Y_j(t) \right) \partial_{x_i}\big|_{\gamma(t)}. \end{align*} This is the coordinate formula forced by (i) and (ii).[/step]

[guided]Let us see the logic that makes this a forcing argument rather than a definition. We started from the assumption that some operator $\nabla_{\partial_t}$ exists with properties (i) and (ii). We then expanded $Y(t) = \sum_j Y_j(t)\,\partial_{x_j}|_{\gamma(t)}$ and applied the properties in turn: - **Leibniz (i)** on the product $Y_j(t) \cdot \partial_{x_j}|_{\gamma(t)}$ splits the derivative into "$Y_j'(t)\,\partial_{x_j}|_{\gamma(t)}$ plus $Y_j(t)\,\nabla_{\partial_t}(\partial_{x_j} \circ \gamma)$". We are using (i) with $f(t) = Y_j(t)$ and with the section being $\partial_{x_j} \circ \gamma$. - **Consistency (ii)** applied to the section $\partial_{x_j} \circ \gamma$, which agrees with the global-on-$U$ vector field $\partial_{x_j}$, converts $\nabla_{\partial_t}(\partial_{x_j} \circ \gamma)|_t$ into $\nabla^{\mathrm{aff}}_{\gamma'(t)} \partial_{x_j}|_{\gamma(t)}$. This is the ambient affine connection evaluated at a tangent vector. - Finally, $\gamma'(t)$ in coordinates is $\sum_k x_k'(t)\,\partial_{x_k}|_{\gamma(t)}$, and $\nabla^{\mathrm{aff}}_{\partial_{x_k}}\partial_{x_j}$ is what the Christoffel symbols define: $\sum_i \Gamma^i_{jk}\,\partial_{x_i}$. Plugging in yields the closed-form expression for the $i$-th component of $\nabla_{\partial_t} Y$. Since each deduction was forced — every "=" above is mandatory given (i) and (ii) and the expansion of $Y$ — any two operators satisfying (i) and (ii) must produce the same coordinate formula on $\gamma^{-1}(U)$. This is the uniqueness conclusion. Why is this uniqueness on a chart enough for global uniqueness? Because $M$ is covered by charts and any point of $(a, b)$ has a chart-neighbourhood around $\gamma(t_0)$; two operators that agree on each $\gamma^{-1}(U)$ agree everywhere on $(a, b)$.[/guided]

[step:Establish uniqueness globally from the coordinate formula] Suppose $\nabla_{\partial_t}$ and $\widetilde{\nabla}_{\partial_t}$ both satisfy (i) and (ii). By the previous step, for every $t_0 \in (a, b)$ there is a chart neighbourhood on which both operators are given by the same coordinate formula in terms of $(Y_j, x_k, \Gamma^i_{jk})$, so \begin{align*} \nabla_{\partial_t} Y\big|_t = \widetilde{\nabla}_{\partial_t} Y\big|_t \end{align*} on $\gamma^{-1}(U)$. Since $(a, b)$ is covered by such intervals, $\nabla_{\partial_t} Y = \widetilde{\nabla}_{\partial_t} Y$ on $(a, b)$. As $Y \in \Gamma(\gamma^* TM)$ was arbitrary, $\nabla_{\partial_t} = \widetilde{\nabla}_{\partial_t}$. This proves uniqueness. [/step]

[step:Define the operator locally via the coordinate formula and verify chart independence]For existence, we define $\nabla_{\partial_t}$ on each $\gamma^{-1}(U)$ by the formula derived above, and then verify the definition is independent of the chart. Let $(U, \varphi)$ and $(\widetilde U, \widetilde \varphi)$ be two charts with $\gamma(t_0) \in U \cap \widetilde U$, with coordinates $(x_1, \dots, x_n)$ and $(\widetilde x_1, \dots, \widetilde x_n)$ respectively. Write the transition $\widetilde x_i = \widetilde x_i(x_1, \dots, x_n)$ on $U \cap \widetilde U$ and denote its Jacobian entries $J^i_j = \partial \widetilde x_i / \partial x_j$. The transformation rule for the coordinate basis, \begin{align*} \partial_{x_j}\big|_p = \sum_{i=1}^n J^i_j(p)\, \partial_{\widetilde x_i}\big|_p, \qquad p \in U \cap \widetilde U, \end{align*} gives the relation $\widetilde Y_i(t) = \sum_j J^i_j(\gamma(t))\, Y_j(t)$ between the component functions of $Y$ in the two charts. The standard [Christoffel transformation law](/theorems/???) for an affine connection reads \begin{align*} \widetilde \Gamma^i_{jk} = \sum_{p, q, r} J^i_p\, (J^{-1})^q_j\, (J^{-1})^r_k\, \Gamma^p_{qr} - \sum_{p, q} J^i_p\, (J^{-1})^q_j\, \frac{\partial (J^{-1})^p_k}{\partial \widetilde x_q} \end{align*} on $U \cap \widetilde U$ (the second term is the non-tensorial correction). A direct — if tedious — calculation, substituting $\widetilde Y_i$, $\widetilde x_k$, and $\widetilde \Gamma^i_{jk}$ into the coordinate formula, shows that the $i$-th component computed in the $\widetilde x$-chart equals $\sum_j J^i_j\, [\text{$j$-th component in $x$-chart}]$. That is, the two coordinate formulas define the same element of $T_{\gamma(t)}M$. Hence the locally defined operator is chart-independent, and the collection of local definitions glues to a global operator $\nabla_{\partial_t}: \Gamma(\gamma^* TM) \to \Gamma(\gamma^* TM)$.[/step]

[guided]The chart-independence check is the only place where the non-tensorial second term in the Christoffel transformation law plays a role. Here is the structure of the computation, without grinding through every index. The coordinate formula has two parts: \begin{align*} \text{(A)} \quad \sum_i Y_i'(t)\, \partial_{x_i}\big|_{\gamma(t)}, \qquad \text{(B)} \quad \sum_{i, j, k} \Gamma^i_{jk}(\gamma(t))\, x_k'(t)\, Y_j(t)\, \partial_{x_i}\big|_{\gamma(t)}. \end{align*} Part (A), the "derivative of components" term, does not transform covariantly by itself: differentiating $\widetilde Y_i(t) = \sum_j J^i_j(\gamma(t))\, Y_j(t)$ produces a cross term $\sum_j \frac{d}{dt}[J^i_j(\gamma(t))]\, Y_j(t)$, which contains $\partial J^i_j / \partial x_k \cdot x_k'(t)$. Part (B), the "Christoffel correction" term, also does not transform covariantly: the non-tensorial piece of the Christoffel transformation law introduces a term involving $\partial J/\partial x$. The miracle — well known in Riemannian geometry — is that these two non-covariant pieces **cancel exactly**, leaving the sum $\text{(A)} + \text{(B)}$ transforming as a vector. This is in fact the original motivation for the Christoffel transformation law: the correction term is defined so that this cancellation works. Rather than reproducing the algebra, we invoke the standard result: the formula $Y_i' + \sum \Gamma^i_{jk}\,x_k'\,Y_j$ is the component expression of a well-defined vector in $T_{\gamma(t)}M$, with no chart dependence. See any text on Riemannian geometry for the detailed verification (Lee's *Riemannian Manifolds* Prop. 4.16, do Carmo §2, or Kobayashi–Nomizu §III.1). Once chart-independence is established, the local operators glue to a global $\nabla_{\partial_t}$ on $\gamma^* TM$.[/guided]

[step:Verify that the constructed operator satisfies (i) and (ii)]We check (i). Let $f \in C^\infty(a, b)$ and $Y \in \Gamma(\gamma^* TM)$. In a chart, $(fY)_j(t) = f(t)\, Y_j(t)$, so $(fY)_j'(t) = f'(t)\, Y_j(t) + f(t)\, Y_j'(t)$. Substituting into the coordinate formula, \begin{align*} \nabla_{\partial_t}(fY)\big|_t &= \sum_i \left( (fY)_i'(t) + \sum_{j,k} \Gamma^i_{jk}(\gamma(t))\, x_k'(t)\, (fY)_j(t) \right) \partial_{x_i}\big|_{\gamma(t)} \\ &= \sum_i \left( f'(t)\, Y_i(t) + f(t) \left[ Y_i'(t) + \sum_{j,k} \Gamma^i_{jk}(\gamma(t))\, x_k'(t)\, Y_j(t) \right] \right) \partial_{x_i}\big|_{\gamma(t)} \\ &= f'(t)\, Y\big|_t + f(t)\, \nabla_{\partial_t} Y\big|_t, \end{align*} so (i) holds. We check (ii). Suppose $Y$ agrees near $t_0$ with $X \circ \gamma$ for a global vector field $X \in \Gamma(TM)$. Write $X = \sum_j X_j\, \partial_{x_j}$ in our chart, with $X_j \in C^\infty(U)$. Then $Y_j(t) = X_j(\gamma(t))$ near $t_0$. Differentiating via the chain rule, \begin{align*} Y_j'(t_0) = \sum_k \frac{\partial X_j}{\partial x_k}(\gamma(t_0))\, x_k'(t_0). \end{align*} Substituting into the coordinate formula at $t = t_0$: \begin{align*} \nabla_{\partial_t} Y\big|_{t_0} = \sum_i \left( \sum_k \frac{\partial X_i}{\partial x_k}(\gamma(t_0))\, x_k'(t_0) + \sum_{j,k} \Gamma^i_{jk}(\gamma(t_0))\, x_k'(t_0)\, X_j(\gamma(t_0)) \right) \partial_{x_i}\big|_{\gamma(t_0)}. \end{align*} The right-hand side is precisely the coordinate expression for $\nabla^{\mathrm{aff}}_{\gamma'(t_0)} X\big|_{\gamma(t_0)}$: indeed, writing $\gamma'(t_0) = \sum_k x_k'(t_0)\, \partial_{x_k}\big|_{\gamma(t_0)}$, \begin{align*} \nabla^{\mathrm{aff}}_{\gamma'(t_0)} X &= \sum_k x_k'(t_0)\, \nabla^{\mathrm{aff}}_{\partial_{x_k}} \left( \sum_j X_j\, \partial_{x_j} \right) \\ &= \sum_{i,k} x_k'(t_0)\, \frac{\partial X_i}{\partial x_k}(\gamma(t_0))\, \partial_{x_i} + \sum_{i,j,k} x_k'(t_0)\, X_j(\gamma(t_0))\, \Gamma^i_{jk}(\gamma(t_0))\, \partial_{x_i}, \end{align*} which matches. Hence (ii) holds. Combining existence and uniqueness, the operator $\nabla_{\partial_t}$ exists and is unique, completing the proof.[/step]

[guided]The verification of (i) is straightforward because the coordinate formula is linear in the component functions and the derivative $\partial_t$ obeys the product rule — we just unfold $(fY)_j' = f'Y_j + fY_j'$ and collect terms. The verification of (ii) uses a single application of the chain rule. The key point is that if $Y$ comes from a global $X$, then $Y_j(t) = X_j(\gamma(t))$ and the derivative $Y_j'(t)$ splits into a $\nabla^{\mathrm{aff}}$-like expression after using the chain rule on $X_j \circ \gamma$: \begin{align*} \frac{d}{dt}[X_j \circ \gamma](t) = \sum_k \frac{\partial X_j}{\partial x_k}(\gamma(t))\, x_k'(t). \end{align*} This is exactly the "derivative of $X$ in the direction $\gamma'(t)$" combined with the Christoffel correction to produce $\nabla^{\mathrm{aff}}_{\gamma'(t)} X$. A small design comment: notice that property (ii) forces the Christoffel correction to appear in the coordinate formula. If we had only required (i), the formula $\nabla_{\partial_t} Y|_t = \sum_i Y_i'(t)\,\partial_{x_i}|_{\gamma(t)}$ (without the Christoffel term) would satisfy (i) — but it would not be chart-independent, and it would fail (ii) for any non-trivial $X$. So the two properties are genuinely working together: (i) localises the problem to coefficients, and (ii) pins down how the basis fields are differentiated.[/guided]