[proofplan] The strategy is to manufacture a smooth vector field $X \in \Gamma(TM)$ that is transverse to $\partial M$ and points into the interior of $M$ at each boundary point, then to flow $\partial M$ along $X$ for short time. We build $X$ locally inside boundary charts (where it can be taken to be the coordinate vector field in the inward normal direction) and glue with a smooth [partition of unity](/page/Partition%20of%20Unity). The flow map $\Phi(p,t) := \varphi^X(t,p)$ is smooth, and at every $(p,0)$ its differential is a linear isomorphism because $X(p)$ is transverse to $T_p \partial M$. The [Inverse Function Theorem](/page/Inverse%20Function%20Theorem) then shows $\Phi$ is a local diffeomorphism near $\partial M \times \{0\}$; compactness of $\partial M$ upgrades this to a uniform $\varepsilon > 0$, and a further shrinking argument secures global injectivity. [/proofplan]

[step:Cover $\partial M$ by boundary-adapted charts and define inward coordinate vector fields]Write $\mathbb{H}^n := \mathbb{R}^{n-1} \times [0, \infty)$ for the closed upper half-space, with coordinates $(x_1, \dots, x_n)$ and boundary $\partial \mathbb{H}^n = \mathbb{R}^{n-1} \times \{0\}$. By the definition of a smooth manifold with boundary, for every $p \in \partial M$ there exists a smooth chart $(V_p, \psi_p)$ with $p \in V_p$ and \begin{align*} \psi_p : V_p &\longrightarrow \psi_p(V_p) \subseteq \mathbb{H}^n \end{align*} satisfying $\psi_p(V_p \cap \partial M) = \psi_p(V_p) \cap \partial \mathbb{H}^n$. Define the local vector field $X_p \in \Gamma(TV_p)$ to be the pullback of $\partial_{x_n}$ on $\mathbb{H}^n$ under $\psi_p$: \begin{align*} X_p(q) := (d\psi_q)^{-1}\!\left(\partial_{x_n}\big|_{\psi_p(q)}\right), \qquad q \in V_p. \end{align*} Each $X_p$ is smooth on $V_p$. By construction, for every $q \in V_p \cap \partial M$ the vector $X_p(q) \in T_qM$ is **inward-pointing**, meaning $(d\psi_p)_q\bigl(X_p(q)\bigr) = \partial_{x_n}$ has strictly positive $x_n$-component.[/step]

[guided]The proof begins by manufacturing the geometric object that will drive everything: a smooth vector field on $M$ that points into the interior at every boundary point. We first build such vector fields locally in boundary charts. A boundary chart at $p \in \partial M$ is a homeomorphism $\psi_p$ from an open neighbourhood $V_p \subseteq M$ of $p$ onto an open subset of the closed half-space $\mathbb{H}^n = \mathbb{R}^{n-1} \times [0,\infty)$, sending $V_p \cap \partial M$ into $\partial \mathbb{H}^n = \mathbb{R}^{n-1} \times \{0\}$. Such charts exist by the very definition of a smooth manifold with boundary. In half-space coordinates the obvious inward-pointing direction is $\partial_{x_n}$ — moving in the positive $x_n$-direction takes a boundary point into the interior. We pull this back to $V_p$ via $\psi_p$: \begin{align*} X_p : V_p &\longrightarrow TV_p \\ q &\longmapsto (d\psi_q)^{-1}\!\bigl(\partial_{x_n}\big|_{\psi_p(q)}\bigr). \end{align*} This is a smooth section of $TV_p$ because $\psi_p$ is a diffeomorphism onto its image and $\partial_{x_n}$ is a smooth coordinate field. By "inward-pointing at $q \in \partial M$" we mean that $X_p(q)$ is not tangent to $\partial M$ at $q$ and, when expressed in any boundary chart at $q$, has positive $x_n$-component. The construction makes this automatic at $q \in V_p \cap \partial M$: in the chart $\psi_p$ itself, $(d\psi_p)_q(X_p(q)) = \partial_{x_n}$, whose $x_n$-component equals $1$. The notion of "inward-pointing" is in fact independent of the choice of boundary chart, since transition maps between boundary charts of $M$ preserve $\partial \mathbb{H}^n$ and hence have Jacobians whose last column has positive $n$-th entry on $\partial \mathbb{H}^n$.[/guided]

[step:Patch the local fields into a global smooth vector field $X$ that is inward-pointing on $\partial M$]Since $M$ is compact, $\partial M$ is closed in $M$ and hence compact. Extract a finite subcover: $\partial M \subseteq V_{p_1} \cup \cdots \cup V_{p_k}$. Set $V_0 := M \setminus \partial M$, which is open in $M$, and let $X_0 := 0 \in \Gamma(TV_0)$. The collection $\mathcal{V} := \{V_0, V_{p_1}, \dots, V_{p_k}\}$ is an open cover of $M$. By the [Existence of Smooth Partitions of Unity](/theorems/57) applied to the smooth manifold $M$ (with boundary) and the open cover $\mathcal{V}$, there exists a smooth [partition of unity](/page/Partition%20of%20Unity) $\{\rho_0, \rho_1, \dots, \rho_k\}$ subordinate to $\mathcal{V}$: each $\rho_i \in C^\infty(M)$ satisfies $0 \le \rho_i \le 1$, $\operatorname{supp}\rho_i \subseteq V_{p_i}$ (with the convention $V_{p_0} := V_0$), and $\sum_{i=0}^k \rho_i \equiv 1$ on $M$. Define \begin{align*} X : M &\longrightarrow TM \\ q &\longmapsto \sum_{i=1}^{k} \rho_i(q)\, X_{p_i}(q), \end{align*} where each summand $\rho_i X_{p_i}$ is extended by zero to all of $M$ (this extension is smooth because $\rho_i$ vanishes outside $V_{p_i}$). Then $X \in \Gamma(TM)$. For any $q \in \partial M$, the term $i = 0$ contributes nothing because $\operatorname{supp}\rho_0 \subseteq V_0 = M \setminus \partial M$, so $\rho_0(q) = 0$. Hence $\sum_{i=1}^k \rho_i(q) = 1$, and at least one $\rho_i(q) > 0$. Pick any boundary chart $(V_{p_j}, \psi_{p_j})$ containing $q$. The $x_n$-component of $(d\psi_{p_j})_q(X_{p_i}(q))$ is positive whenever $q \in V_{p_i}$, by the chart-change observation in the previous step. Thus \begin{align*} (d\psi_{p_j})_q(X(q)) = \sum_{i : q \in V_{p_i}} \rho_i(q)\, (d\psi_{p_j})_q(X_{p_i}(q)) \end{align*} has strictly positive $x_n$-component. Consequently $X(q)$ is inward-pointing, and in particular $X(q) \notin T_q \partial M$.[/step]

[guided]We now stitch the local vector fields $X_{p_i}$ into a single global smooth vector field $X$ on $M$ using a [partition of unity](/page/Partition%20of%20Unity), in such a way that the inward-pointing property survives the gluing. Compactness of $M$ makes $\partial M$ a closed subset of a [compact space](/page/Compact%20Space), hence compact, so we may extract a finite subcover $V_{p_1}, \dots, V_{p_k}$ of $\partial M$ from the family $\{V_p : p \in \partial M\}$. To extend to an open cover of all of $M$ we add the interior $V_0 := M \setminus \partial M$ (open because $\partial M$ is closed) with the zero vector field $X_0 = 0$. The [Existence of Smooth Partitions of Unity](/theorems/57) requires the manifold to be smooth (which $M$ is, as a manifold with boundary admitting smooth partitions of unity by the standard construction extending the boundaryless case) and the cover to be open (which $\mathcal{V}$ is). It produces functions $\rho_i \in C^\infty(M)$, $0 \le \rho_i \le 1$, with $\operatorname{supp}\rho_i \subseteq V_{p_i}$ and $\sum_i \rho_i \equiv 1$ on $M$. We then form \begin{align*} X(q) = \sum_{i=1}^k \rho_i(q)\, X_{p_i}(q), \end{align*} omitting the $i = 0$ term (which would be $\rho_0 \cdot 0 = 0$ anyway). Each summand $\rho_i X_{p_i}$ extends smoothly by zero outside $V_{p_i}$ because $\rho_i$ vanishes there, and a finite sum of smooth sections is smooth. So $X \in \Gamma(TM)$. Why does $X$ remain inward-pointing on $\partial M$? Three facts conspire. First, $\rho_0(q) = 0$ at any $q \in \partial M$ because $\operatorname{supp}\rho_0 \subseteq V_0$ does not meet $\partial M$. So the [partition of unity](/page/Partition%20of%20Unity) reduces on $\partial M$ to a convex combination of the boundary-chart fields. Second, fixing any boundary chart $\psi_{p_j}$ at $q$, each $(d\psi_{p_j})_q(X_{p_i}(q))$ for $q \in V_{p_i}$ has positive $x_n$-component — this uses that the transition map $\psi_{p_j} \circ \psi_{p_i}^{-1}$ is a diffeomorphism of half-space neighbourhoods preserving $\partial \mathbb{H}^n$, forcing its Jacobian at boundary points to have positive entry in row $n$, column $n$. Third, a positive convex combination of vectors with strictly positive $x_n$-components has strictly positive $x_n$-component. The conclusion is that for every $q \in \partial M$, the vector $X(q) \in T_qM$ is inward-pointing, in particular transverse to $T_q\partial M$: \begin{align*} T_qM = T_q\partial M \oplus \mathbb{R} \cdot X(q). \end{align*} This transversality is the geometric input that makes the rest of the proof work.[/guided]

[step:Define the candidate collar map $\Phi$ via the flow of $X$]For each $p \in \partial M$, by the [Picard–Lindelöf Theorem](/theorems/69) applied to the smooth ODE $\dot{\gamma}(t) = X(\gamma(t))$ in any boundary chart at $p$, there exist an open neighbourhood $V$ of $p$ in $M$ and a number $\delta > 0$ such that for every $q \in V$ the maximal solution $\gamma_q : J_q \to M$ of \begin{align*} \dot{\gamma}_q(t) = X(\gamma_q(t)), \qquad \gamma_q(0) = q, \end{align*} is defined on an interval $J_q \supseteq [0, \delta)$ (forward time suffices since the inward-pointing direction pushes boundary trajectories into the interior, where flow extension is standard). The map $(q, t) \mapsto \gamma_q(t)$ is smooth on $V \times [0, \delta)$. Covering the compact set $\partial M$ by finitely many such neighbourhoods $V_1, \dots, V_N$ with corresponding times $\delta_1, \dots, \delta_N$ and setting $\delta_* := \min_i \delta_i > 0$, we obtain a smooth map \begin{align*} \Phi : \partial M \times [0, \delta_*) &\longrightarrow M \\ (p, t) &\longmapsto \gamma_p(t). \end{align*} By construction $\Phi(p, 0) = \gamma_p(0) = p$ for every $p \in \partial M$.[/step]

[guided]Having built $X$, we now integrate it. The flow of $X$ starting at a boundary point $p$ is the unique curve $\gamma_p : J_p \to M$ with $\gamma_p(0) = p$ and $\dot{\gamma}_p(t) = X(\gamma_p(t))$. Standard ODE theory ([Picard–Lindelöf](/theorems/69)) provides local existence, uniqueness, and smooth dependence on initial data for the system $\dot\gamma = X(\gamma)$. We invoke it inside a boundary chart $(V_{p_j}, \psi_{p_j})$ at each $p \in \partial M$: in coordinates the ODE becomes $\dot y(t) = (\psi_{p_j})_* X (y(t))$ on the smooth [open set](/page/Open%20Set) $\psi_{p_j}(V_{p_j}) \subseteq \mathbb{H}^n$, with smooth right-hand side. Picard–Lindelöf yields a smooth map $(q, t) \mapsto y(t)$ defined for $q$ in a neighbourhood of $\psi_{p_j}(p)$ and $t$ in a small interval $(-\delta, \delta)$. One subtlety: we are working on a manifold with boundary, so for $t < 0$ the trajectory might try to leave $\mathbb{H}^n$ through $\partial \mathbb{H}^n$. We only need $t \ge 0$ — and for $t \ge 0$ the inward-pointing property of $X$ on $\partial M$ guarantees that trajectories starting at boundary points move immediately into the interior, where they extend by standard flow theory in the boundaryless interior. Covering compact $\partial M$ by finitely many such local neighbourhoods $V_1, \dots, V_N$ and taking the minimum of the local existence times $\delta_i$, we patch together a globally defined smooth map \begin{align*} \Phi : \partial M \times [0, \delta_*) &\longrightarrow M, \qquad \Phi(p, t) := \gamma_p(t). \end{align*} The boundary condition $\Phi(p, 0) = p$ holds by construction of the flow. The remaining work is to show $\Phi$ is a diffeomorphism onto an [open set](/page/Open%20Set) near $\partial M$ for some sufficiently small $\varepsilon \le \delta_*$.[/guided]

[step:Show $d\Phi_{(p,0)}$ is a linear isomorphism for every $p \in \partial M$] Fix $p \in \partial M$. The domain of $\Phi$ near $(p, 0)$ is locally modelled on $T_p \partial M \times \mathbb{R}_{\ge 0}$, with formal tangent space $T_{(p,0)}(\partial M \times [0,\delta_*)) \cong T_p \partial M \oplus \mathbb{R}$. We compute $d\Phi_{(p,0)}$ on each factor. For $v \in T_p \partial M$, the curve $s \mapsto (p_s, 0)$ in $\partial M \times \{0\}$ with $p_0 = p$ and $\dot p_0 = v$ satisfies $\Phi(p_s, 0) = p_s$, so \begin{align*} d\Phi_{(p,0)}(v, 0) = \frac{d}{ds}\bigg|_{s=0} p_s = v \in T_p M. \end{align*} For the time direction, the curve $t \mapsto (p, t)$ has image $\gamma_p(t)$ under $\Phi$, so \begin{align*} d\Phi_{(p,0)}(0, 1) = \frac{d}{dt}\bigg|_{t=0} \gamma_p(t) = X(p) \in T_p M. \end{align*} By linearity, $d\Phi_{(p,0)}(v, a) = v + a X(p)$. Since $X(p)$ is inward-pointing, in particular $X(p) \notin T_p \partial M$, the decomposition $T_pM = T_p\partial M \oplus \mathbb{R} \cdot X(p)$ is a direct sum (both subspaces of dimensions $n-1$ and $1$ sum to dimension $n$). The map $(v, a) \mapsto v + aX(p)$ is therefore a linear isomorphism $T_p\partial M \oplus \mathbb{R} \to T_pM$. [/step]

[step:Apply the Inverse Function Theorem and compactness to obtain uniform local diffeomorphism]By the [Inverse Function Theorem](/theorems/51), at each $(p, 0) \in \partial M \times \{0\}$ the smooth map $\Phi$ — viewed locally as a map between open subsets of $n$-manifolds via boundary charts on the source and target — restricts to a diffeomorphism on some open neighbourhood. Concretely, there exist an open neighbourhood $W_p \subseteq \partial M \times [0, \delta_*)$ of $(p, 0)$ and an open neighbourhood $U_p \subseteq M$ of $p$ such that \begin{align*} \Phi\big|_{W_p} : W_p \longrightarrow U_p \end{align*} is a diffeomorphism. We may assume $W_p$ has the product form $W_p = O_p \times [0, \eta_p)$ for an open $O_p \subseteq \partial M$ containing $p$ and $\eta_p \in (0, \delta_*)$, by shrinking $W_p$. The family $\{O_p : p \in \partial M\}$ covers the compact set $\partial M$; extract a finite subcover $\partial M \subseteq O_{p_1} \cup \cdots \cup O_{p_m}$ and set \begin{align*} \varepsilon_1 := \min_{1 \le j \le m} \eta_{p_j} > 0. \end{align*} Then $\partial M \times [0, \varepsilon_1) \subseteq \bigcup_{j=1}^m W_{p_j}$, and on each $W_{p_j}$ the map $\Phi$ is a local diffeomorphism. Hence $\Phi$ is a smooth local diffeomorphism on the entire set $\partial M \times [0, \varepsilon_1)$.[/step]

[guided]Having shown that the differential $d\Phi_{(p,0)}$ is an isomorphism for every $p \in \partial M$, we invoke the [Inverse Function Theorem](/theorems/51) to upgrade pointwise invertibility of the differential into local invertibility of the map itself. The [Inverse Function Theorem](/theorems/51) requires a smooth map between (open subsets of) manifolds of equal dimension whose differential at a point is a linear isomorphism. We verify each ingredient. Both source and target are $n$-dimensional smooth manifolds (the source has boundary, but the theorem applies equally on the half-space side using boundary charts). Smoothness of $\Phi$ was established in the previous step. Bijectivity of $d\Phi_{(p,0)}$ was just shown. The conclusion: $\Phi$ restricts to a diffeomorphism from some open neighbourhood $W_p$ of $(p, 0)$ onto some open neighbourhood $U_p$ of $\Phi(p, 0) = p$ in $M$. By shrinking $W_p$ inside $\partial M \times [0, \delta_*)$, we may take it of product form $O_p \times [0, \eta_p)$ — a small open piece of $\partial M$ times a short time interval. This product form is convenient for the compactness argument. Now compactness of $\partial M$ enters. The slices $\{O_p\}_{p \in \partial M}$ form an open cover of $\partial M$; finitely many suffice — $O_{p_1}, \dots, O_{p_m}$. Setting $\varepsilon_1 := \min_j \eta_{p_j} > 0$, every $(p, t) \in \partial M \times [0, \varepsilon_1)$ lies in some $W_{p_j} = O_{p_j} \times [0, \eta_{p_j})$. Hence $\Phi$ is locally diffeomorphic at every point of $\partial M \times [0, \varepsilon_1)$. A local diffeomorphism is **not** automatically a diffeomorphism — it can fail to be injective globally (e.g. $\mathbb{R} \to S^1$, $t \mapsto e^{it}$). The next step removes that obstruction.[/guided]

[step:Shrink $\varepsilon$ to enforce global injectivity]We claim that for some $0 < \varepsilon \le \varepsilon_1$, the map $\Phi|_{\partial M \times [0, \varepsilon)}$ is injective. Suppose, for contradiction, that no such $\varepsilon$ exists. Then for every $k \in \mathbb{N}$ there exist distinct points \begin{align*} (p_k, s_k),\, (q_k, t_k) \in \partial M \times [0, \tfrac{1}{k}) \quad \text{with} \quad \Phi(p_k, s_k) = \Phi(q_k, t_k). \end{align*} By compactness of $\partial M$, pass to a subsequence (still denoted $k$) such that $p_k \to p^* \in \partial M$ and $q_k \to q^* \in \partial M$. Since $s_k, t_k \in [0, 1/k)$, we have $s_k \to 0$ and $t_k \to 0$. Continuity of $\Phi$ gives \begin{align*} \Phi(p^*, 0) = \lim_{k \to \infty} \Phi(p_k, s_k) = \lim_{k \to \infty} \Phi(q_k, t_k) = \Phi(q^*, 0), \end{align*} i.e. $p^* = q^*$ (since $\Phi(\cdot, 0) = \mathrm{id}_{\partial M}$). By the previous step, $\Phi$ is a diffeomorphism from some open neighbourhood $W_{p^*}$ of $(p^*, 0)$ onto its image. For all sufficiently large $k$, both $(p_k, s_k)$ and $(q_k, t_k)$ lie in $W_{p^*}$ (they converge to $(p^*, 0)$), yet $\Phi$ identifies them — contradicting injectivity of $\Phi|_{W_{p^*}}$. Fix such an $\varepsilon \in (0, \varepsilon_1]$ from now on.[/step]

[guided]A local diffeomorphism between manifolds of equal dimension is open and a homeomorphism onto its image precisely when it is also injective. We have $\Phi$ a local diffeomorphism on $\partial M \times [0, \varepsilon_1)$; we now find $\varepsilon \in (0, \varepsilon_1]$ on which it is globally injective. The argument is a standard compactness-plus-continuity contradiction. If injectivity failed for every $\varepsilon > 0$, we could pick witnesses $(p_k, s_k) \ne (q_k, t_k)$ with $\Phi(p_k, s_k) = \Phi(q_k, t_k)$ and $s_k, t_k < 1/k$. Compactness of $\partial M$ extracts convergent subsequences $p_k \to p^*$, $q_k \to q^*$, while the times tend to zero. Continuity of $\Phi$ identifies $\Phi(p^*, 0) = \Phi(q^*, 0)$, and the boundary condition $\Phi(\cdot, 0) = \mathrm{id}$ forces $p^* = q^*$. The contradiction comes from locality: near $(p^*, 0)$ the map $\Phi$ is a diffeomorphism onto its image, hence injective on some open neighbourhood $W_{p^*}$. The two sequences $(p_k, s_k)$ and $(q_k, t_k)$ both converge to $(p^*, 0)$, so eventually both lie in $W_{p^*}$, where $\Phi$ cannot identify them. This contradicts the assumption that they were chosen to have equal $\Phi$-images. Hence some $\varepsilon \in (0, \varepsilon_1]$ makes $\Phi|_{\partial M \times [0, \varepsilon)}$ injective. From this point on we fix that $\varepsilon$.[/guided]

[step:Conclude that $\Phi$ is a diffeomorphism onto an open neighbourhood of $\partial M$] Let $U := \Phi(\partial M \times [0, \varepsilon)) \subseteq M$. Since $\Phi$ is a local diffeomorphism on $\partial M \times [0, \varepsilon)$, it is an open map; in particular $U$ is open in $M$. The set $U$ contains $\partial M$ because $\Phi(p, 0) = p$ for every $p \in \partial M$. By the previous step, $\Phi|_{\partial M \times [0, \varepsilon)} : \partial M \times [0, \varepsilon) \to U$ is a smooth bijection that is, locally at every point, a diffeomorphism. A smooth bijection between manifolds of equal dimension which is everywhere a local diffeomorphism is a diffeomorphism, since its global inverse is locally the composition of local smooth inverses and hence smooth. Therefore \begin{align*} \Phi : \partial M \times [0, \varepsilon) \longrightarrow U \end{align*} is a diffeomorphism satisfying $\Phi(p, 0) = p$ for all $p \in \partial M$, which is the conclusion of the theorem. [/step]