[proofplan] Work in a chart around $p$ to reduce to the problem of solving an ODE system $\dot y_i = a_i(y)$ on an open subset of $\mathbb{R}^n$, where the $a_i$ are the smooth coordinate components of $X$. The Picard-Lindelöf theorem with smooth dependence on parameters produces, on a small space-time cylinder, a unique smooth solution $\varphi$ with initial condition $\varphi_0 = \mathrm{id}$. Property (iii) is an immediate consequence of the ODE and the chain rule, while property (ii) follows from uniqueness of solutions because both sides of the group identity satisfy the same initial value problem in the time variable. Pulling the chart-level flow back to $M$ gives the required local flow. [/proofplan] [step:Reduce to coordinate form by choosing a chart at $p$] Choose a smooth chart $(W, \psi)$ on $M$ with $p \in W$ and $\psi(p) = 0 \in \mathbb{R}^n$. Write $\widehat W := \psi(W) \subseteq \mathbb{R}^n$, which is open and contains $0$. The vector field $X$ pushes forward to a smooth vector field on $\widehat W$ whose components we denote by $a_1, \dots, a_n \in C^\infty(\widehat W, \mathbb{R})$, so that for every $y \in \widehat W$, \begin{align*} (\psi_* X)(y) = \sum_{i=1}^{n} a_i(y)\, \partial_{y_i}. \end{align*} Here $\psi_* X$ is defined pointwise by $(\psi_* X)(\psi(x)) = d\psi_x(X(x))$ for $x \in W$. The flow problem for $X$ on $W$ is equivalent, under the chart $\psi$, to the flow problem for $\psi_* X$ on $\widehat W$. Throughout the remainder we work on $\widehat W$, write $y = (y_1, \dots, y_n)$ for coordinates in $\mathbb{R}^n$, and construct a local flow $\widehat\varphi$ for the system \begin{align*} \frac{d y_i}{d t} = a_i(y_1(t), \dots, y_n(t)), \qquad y_i(0) = y_{0, i}, \qquad i = 1, \dots, n, \end{align*} with initial condition $y_0 = (y_{0,1}, \dots, y_{0,n}) \in \widehat W$. At the end, $\varphi := \psi^{-1} \circ \widehat\varphi \circ (\mathrm{id} \times \psi)$ will be the required local flow on $M$. [/step] [step:Apply the Picard-Lindelöf theorem with smooth parameter dependence to solve the ODE system] We apply the [Picard-Lindelöf Theorem with Smooth Dependence on Initial Conditions](/theorems/???) to the vector-valued map \begin{align*} a: \widehat W &\to \mathbb{R}^n \\ y &\mapsto (a_1(y), \dots, a_n(y)). \end{align*} Hypotheses verification: - Open domain: $\widehat W \subseteq \mathbb{R}^n$ is open and contains $0$ (chosen above). - Smoothness of $a$: each $a_i \in C^\infty(\widehat W, \mathbb{R})$ since $X$ is smooth and charts preserve smoothness; hence $a \in C^\infty(\widehat W, \mathbb{R}^n)$. In particular, $a$ is continuously differentiable and, on any compact subset $K \subset \widehat W$, locally Lipschitz with Lipschitz constant $L_K := \sup_K \|Ja\|_{\mathrm{op}} < \infty$. The theorem produces $\varepsilon > 0$, an open neighbourhood $\widehat U \subseteq \widehat W$ of $0$, and a smooth map \begin{align*} \widehat\varphi: (-\varepsilon, \varepsilon) \times \widehat U &\to \widehat W \\ (t, y_0) &\mapsto \widehat\varphi(t, y_0) \end{align*} such that for every $y_0 \in \widehat U$ the curve $t \mapsto \widehat\varphi(t, y_0)$ is the unique $C^1$ solution on $(-\varepsilon, \varepsilon)$ of the initial value problem \begin{align*} \frac{\partial \widehat\varphi}{\partial t}(t, y_0) = a(\widehat\varphi(t, y_0)), \qquad \widehat\varphi(0, y_0) = y_0. \end{align*} Uniqueness is in the class of $C^1$ curves with values in $\widehat W$; since $a$ is smooth, any $C^1$ solution is automatically smooth by bootstrapping the ODE. Smoothness of $\widehat\varphi$ in the joint variable $(t, y_0)$ is part of the conclusion of the smooth-dependence form of Picard-Lindelöf. [/step] [step:Transport $\widehat\varphi$ back to $M$ to define the local flow $\varphi$] Set $U_p := \psi^{-1}(\widehat U) \subseteq W$, an open neighbourhood of $p$ in $M$ with $\psi(p) = 0 \in \widehat U$. Define \begin{align*} \varphi: (-\varepsilon, \varepsilon) \times U_p &\to W \subseteq M \\ (t, x) &\mapsto \psi^{-1}\bigl( \widehat\varphi(t, \psi(x)) \bigr). \end{align*} This is well-defined: for $x \in U_p$ and $|t| < \varepsilon$, $\widehat\varphi(t, \psi(x)) \in \widehat W = \psi(W)$, so $\psi^{-1}$ applies. It is smooth as a composition of smooth maps ($\psi$, $\widehat\varphi$, $\psi^{-1}$). By construction $\varphi(0, x) = \psi^{-1}(\widehat\varphi(0, \psi(x))) = \psi^{-1}(\psi(x)) = x$, so $\varphi_0 = \mathrm{id}_{U_p}$. This establishes property (i): smoothness of $\varphi$ on $(-\varepsilon, \varepsilon) \times U_p$. [/step] [step:Verify property (iii): the infinitesimal generator of $\varphi$ is $X$] Fix $x \in U_p$ and $f \in C^\infty(M)$. Set $\widetilde f := f \circ \psi^{-1}: \widehat W \to \mathbb{R}$, which is smooth. Using the definition of $\varphi$ and the chain rule, \begin{align*} \left. \frac{d}{dt} \right|_{t=0} (f \circ \varphi_t)(x) &= \left. \frac{d}{dt} \right|_{t=0} \widetilde f\bigl( \widehat\varphi(t, \psi(x)) \bigr) \\ &= \sum_{i=1}^{n} \frac{\partial \widetilde f}{\partial y_i}(\psi(x)) \cdot \left. \frac{\partial \widehat\varphi_i}{\partial t} \right|_{t=0, y_0 = \psi(x)} \\ &= \sum_{i=1}^{n} \frac{\partial \widetilde f}{\partial y_i}(\psi(x)) \cdot a_i(\psi(x)), \end{align*} where in the last equality we used the ODE $\partial_t \widehat\varphi = a \circ \widehat\varphi$ evaluated at $(t, y_0) = (0, \psi(x))$ together with $\widehat\varphi(0, \psi(x)) = \psi(x)$. The right-hand side is precisely the expression, in the chart $(W, \psi)$, of $X(x)$ applied to $f$: \begin{align*} X(x)(f) = \sum_{i=1}^{n} a_i(\psi(x))\, \frac{\partial \widetilde f}{\partial y_i}(\psi(x)). \end{align*} Thus $(X(x))(f) = \left. \frac{d}{dt} \right|_{t=0} (f \circ \varphi_t)(x)$, establishing property (iii). [/step] [step:Verify property (ii) by applying ODE uniqueness] Fix $x \in U_p$ and $t \in (-\varepsilon, \varepsilon)$ with $\varphi_t(x) \in U_p$. Consider the two smooth curves $\gamma_1, \gamma_2: J \to M$ defined on the open interval \begin{align*} J := \{s \in \mathbb{R} : s, t + s \in (-\varepsilon, \varepsilon)\} \end{align*} by \begin{align*} \gamma_1(s) &:= \varphi_{t+s}(x), \\ \gamma_2(s) &:= \varphi_s(\varphi_t(x)). \end{align*} Both curves are well-defined: $\gamma_1$ because $t + s \in (-\varepsilon, \varepsilon)$ and $x \in U_p$; $\gamma_2$ because $s \in (-\varepsilon, \varepsilon)$ and $\varphi_t(x) \in U_p$. Initial values: $\gamma_1(0) = \varphi_t(x)$ and $\gamma_2(0) = \varphi_0(\varphi_t(x)) = \varphi_t(x)$, so $\gamma_1(0) = \gamma_2(0)$. Both curves satisfy the same ODE. In the chart $(W, \psi)$, using the chain rule and the ODE for $\widehat\varphi$: \begin{align*} \frac{d}{ds} \psi(\gamma_1(s)) = \frac{d}{ds} \widehat\varphi(t+s, \psi(x)) = a(\widehat\varphi(t+s, \psi(x))) = a(\psi(\gamma_1(s))), \end{align*} and symmetrically \begin{align*} \frac{d}{ds} \psi(\gamma_2(s)) = \frac{d}{ds} \widehat\varphi(s, \psi(\varphi_t(x))) = a(\widehat\varphi(s, \psi(\varphi_t(x)))) = a(\psi(\gamma_2(s))). \end{align*} Hence $\psi \circ \gamma_1$ and $\psi \circ \gamma_2$ are both solutions, on $J$, of the initial value problem \begin{align*} \frac{d \eta}{d s}(s) = a(\eta(s)), \qquad \eta(0) = \psi(\varphi_t(x)) \in \widehat U. \end{align*} By the uniqueness part of the [Picard-Lindelöf Theorem](/theorems/???) (already invoked in the second step), solutions of this IVP in $\widehat W$ are unique on connected neighbourhoods of $0$. Therefore $\psi \circ \gamma_1 = \psi \circ \gamma_2$ on $J$, and applying $\psi^{-1}$ gives $\gamma_1 = \gamma_2$ on $J$. Evaluating at arbitrary $s \in J$ yields \begin{align*} \varphi_{t+s}(x) = \varphi_s(\varphi_t(x)), \end{align*} which is property (ii). [/step] [step:Verify uniqueness of the local flow] Suppose $\varphi'$ is any smooth map $(-\varepsilon', \varepsilon') \times U_p' \to M$ (with $\varepsilon' > 0$ and $U_p' \ni p$ open) satisfying properties (i), (ii), (iii). Shrinking $\varepsilon'$ and $U_p'$ we may assume $\varphi'$ maps into $W$ and $U_p' \subseteq U_p$. For each fixed $x \in U_p'$, property (iii) combined with property (ii) implies that, in chart coordinates, the curve $t \mapsto \psi(\varphi'_t(x))$ satisfies \begin{align*} \frac{d}{dt} \psi(\varphi'_t(x)) = a(\psi(\varphi'_t(x))), \qquad \psi(\varphi'_0(x)) = \psi(x). \end{align*} This is exactly the same IVP satisfied by $t \mapsto \widehat\varphi(t, \psi(x)) = \psi(\varphi_t(x))$. By uniqueness in Picard-Lindelöf, these two solutions coincide on their common interval of definition, so $\varphi'_t(x) = \varphi_t(x)$ for all such $(t, x)$. Thus the local flow is unique up to restriction of domain. [/step] [step:Collect the conclusions] We have constructed $\varepsilon > 0$, an open set $U_p \ni p$, and a smooth map $\varphi: (-\varepsilon, \varepsilon) \times U_p \to M$ that satisfies properties (i), (ii), (iii) in the statement, and we have established uniqueness up to domain restriction. This completes the proof of the existence and uniqueness of local flows for a smooth vector field. [/step]