description/proof of that for \(C^\infty\) manifold with boundary and \(C^\infty\) vectors field over manifold with boundary, integral curves into interior of manifold with boundary that agree at parameters point agree over common parameters area
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of integral curve of \(C^\infty\) vectors field over \(C^\infty\) manifold with boundary.
- The reader admits the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
- The reader admits the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
- The reader admits the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
- The reader admits the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any \(C^\infty\) vectors field over the manifold with boundary, any integral curves into the interior of the manifold with boundary that agree at any parameters point agree over the common parameters area.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\((TM, M, \pi)\): \(= \text{ the tangent vectors bundle over } M\)
\(s\): \(\in \{\text{ the } C^\infty \text{ sections of } \pi\}\)
\(J_1\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(J_2\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(J\): \(= J_1 \cap J_2 \subseteq \mathbb{R}\)
\(\gamma_1\): \(: J_1 \to Int (M)\), \(\in \{\text{ the integral curves of } s\}\)
\(\gamma_2\): \(: J_2 \to Int (M)\), \(\in \{\text{ the integral curves of } s\}\)
//
Statements:
\(\exists r_0 \in J (\gamma_1 (r_0) = \gamma_2 (r_0))\)
\(\implies\)
\(\gamma_1 \vert_J = \gamma_2 \vert_J\)
//
\(Int (M)\) means the manifold interior of \(M\).
2: Note
The reason why this proposition requires \(\gamma_j\) to be into \(Int (M)\) is that otherwise, we would not be able to apply the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, which requires the domain of \(f\) to be an open subset of \(\mathbb{R}^d\): for any boundary point, \(m \in M\), any chart around \(m\), \((U_m \subseteq M, \phi_m)\), and the induced chart, \((\pi^{-1} (U_m) \subseteq TM, \widetilde{\phi_m})\), \(f = \pi_1 \circ \widetilde{\phi_m} \circ s \circ {\phi_m}^{-1}: \phi_m (U_m) \to \mathbb{R}^d\), where \(\pi_1: \widetilde{\phi_m} (\pi^{-1} (U_m)) = \mathbb{R}^d \times \phi_m (U_m) \to \mathbb{R}^d\) is the projection, would not be any map from open subset of \(\mathbb{R}^d\).
3: Proof
Whole Strategy: Step 0: suppose that \(J\) is not any 1 point interval hereafter; Step 1: when \(r_0\) is not in the interior of \(J\), expand \(J_1\) or \(J_2\) (or both) to make \(r_0\) to be in the interior of \(J\); Step 2: suppose that \(J\) is not upper bounded; Step 3: suppose that \(\gamma_1 \vert_{[r_0, \infty)} \neq \gamma_2 \vert_{[r_0, \infty)}\), and find a contradiction; Step 4: suppose that \(J\) is upper bounded; Step 5: suppose that \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\) was not the upper boundary, and find a contradiction; Step 6: suppose that \(J\) is not lower bounded; Step 7: suppose that \(\gamma_1 \vert_{(- \infty, r_0]} \neq \gamma_2 \vert_{(- \infty, r_0]}\), and find a contradiction; Step 8: suppose that \(J\) is lower bounded; Step 9: suppose that \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\) was not the lower boundary, and find a contradiction; Step 10: conclude the proposition.
Step 0:
When \(J\) is any 1 point interval, the proposition obviously holds.
So, let us suppose that \(J\) is not any 1 point interval, hereafter.
Step 1:
When \(r_0\) is not in the interior of \(J\), let us expand \(J_1\) or \(J_2\) (or both) to make \(r_0\) to be in the interior of \(J\).
\(r_0\)'s not in the interior of \(J\) means that \(r_0\) is a closed boundary of \(J_1\) or is a closed boundary of \(J_2\) or both.
Let us suppose that \(r_0\) is a closed boundary of \(J_j\).
Let \(m_0 := \gamma_1 (r_0) = \gamma_2 (r_0)\).
Let us take a interior chart around \(m_0\), \((U_{m_0} \subseteq M, \phi_{m_0})\), and the induced chart, \((\pi^{-1} (U_{m_0}) \subseteq TM, \widetilde{\phi_{m_0}})\).
Let us think of the ordinary differential equation for \(\phi_{m_0} \circ \gamma\), \(\partial_1 (\phi_{m_0} \circ \gamma (r)) = \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{- 1} (\phi_{m_0} \circ \gamma (r))\), where \(\pi_1: \widetilde{\phi_{m_0}} (\pi^{-1} (U_{m_0})) = \mathbb{R}^d \times \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m_0} \circ \gamma (r_0) = \phi_{m_0} (m_0)\).
\(f: \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d = \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{- 1}\) is \(C^\infty\), because \(s\) is \(C^\infty\).
\(f\) is locally Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, there is a \(B_{\phi_{m_0} (m_0), K}\) such that \(\overline{B_{\phi_{m_0} (m_0), K}} \subseteq \phi_{m_0} (U_{m_0})\) and \(f\) is Lipschitz over \(B_{\phi_{m_0} (m_0), K}\) with \(L\).
\(f \le M'\) for an \(M' \in \mathbb{R}\) over \(\overline{B_{\phi_{m_0} (m_0), K}}\), so, over \(B_{\phi_{m_0} (m_0), K}\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
So, by the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area, there is the unique solution for the equation over \([r_0 - \epsilon_1, r_0 + \epsilon_2]\), where \(0 \lt \epsilon_1, \epsilon_2\), \(\phi_{m_0} \circ \gamma: [r_0 - \epsilon_1, r_0 + \epsilon_2] \to B_{\phi_{m_0} (m_0), K}\).
That means that \(d \gamma / d r (r) = s (\gamma (r))\), so, \(\gamma\) is an integral curve of \(s\).
\(\phi_{m_0} \circ \gamma \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]} = \phi_{m_0} \circ \gamma_j \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\), because \(\phi_{m_0} \circ \gamma \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\) and \(\phi_{m_0} \circ \gamma_j \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\) are some solutions of the ordinary differential equation with initial condition while the solution is unique.
That means that \(\gamma \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]} = \gamma_j \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\).
So, \(\gamma_j\) can be extended over \(J_j \cup [r_0 - \epsilon_1, r_0 + \epsilon_2]\) such that it is the original \(\gamma_j\) over \(J_j\) and it is \(\gamma\) over \([r_0 - \epsilon_1, r_0 + \epsilon_2]\).
Let \(J_j \cup [r_0 - \epsilon_1, r_0 + \epsilon_2]\) be denoted as \(J_j\) hereafter.
Now, \(r_0\) is in the interior of \(J_j\).
If the proposition is proved for the wider \(J_j\) s, the proposition is true for the original \(J_j\) s, because the \(J\) for the latter is contained in the \(J\) for the former.
So, hereafter, we suppose that \(r_0\) is in the interior of \(J\).
Step 2:
\(J\) may be not upper bounded or be upper bounded.
Let us suppose that \(J\) is not upper bounded.
Step 3:
Let us suppose that \(\gamma_1 \vert_{[r_0, \infty)} \neq \gamma_2 \vert_{[r_0, \infty)}\).
Let us define \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\), where the supremum was taken in \(\mathbb{R}\), which would exist as \(r' \in \mathbb{R}\), by the supposition.
\(r' \in Int (J)\), obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_0, \infty)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' - \delta / 2) \neq \gamma_2 (r' - \delta / 2)\), a contradiction against that \(r'\) was the supremum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).
\(f\) is locally Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r_0, r' + \delta / 2]} = \gamma_2 \vert_{[r_0, r' + \delta / 2]}\), a contradiction against that \(r'\) was the supremum.
So, \(\gamma_1 \vert_{[r_0, \infty)} = \gamma_2 \vert_{[r_0, \infty)}\).
Step 4:
Let us suppose that \(J\) is upper bounded with the upper boundary, \(r_2\).
Step 5:
Note that \(J\) may be upper open or upper closed.
What we will do is parallel to the case that \(J\) is not upper bounded.
Let us define \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\), where the supremum is taken in \(\mathbb{R}\), which exists as \(r' \in \mathbb{R}\), because \(J\) is upper bounded.
Let us suppose that \(r'\) was not the upper boundary, which would mean \(r' \lt r_2\).
\(r' \in Int (J)\) whether \(J\) is upper open or upper closed, obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_0, r_2)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' - \delta / 2) \neq \gamma_2 (r' - \delta / 2)\), a contradiction against that \(r'\) was the supremum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).
\(f\) is locally Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r_0, r' + \delta / 2]} = \gamma_2 \vert_{[r_0, r' + \delta / 2]}\), a contradiction against that \(r'\) was the supremum.
So, \(r'\) is the upper boundary, which means \(r' = r_2\).
That means that \(\gamma_1 \vert_{[r_0, r_2)} = \gamma_2 \vert_{[r_0, r_2)}\) or \(\gamma_1 \vert_{[r_0, r_2]} = \gamma_2 \vert_{[r_0, r_2]}\) according to whether \(J\) is upper open or is upper closed: \(\gamma_1 (r_2) = \gamma_2 (r_2)\) can be proved as before.
Step 6:
For the area smaller than \(r_0\), the logic is parallel.
\(J\) may be not lower bounded or be lower bounded.
Let us suppose that \(J\) is not lower bounded.
Step 7:
Let us suppose that \(\gamma_1 \vert_{(- \infty, r_0]} \neq \gamma_2 \vert_{(- \infty, r_0]}\).
Let us define \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\), where the infimum was taken in \(\mathbb{R}\), which would exist as \(r' \in \mathbb{R}\), by the supposition.
\(r' \in Int (J)\), obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (- \infty, r_0)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' + \delta / 2) \neq \gamma_2 (r' + \delta / 2)\), a contradiction against that \(r'\) was the infimum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).
\(f\) is locally Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r' - \delta / 2, r_0]} = \gamma_2 \vert_{[r' - \delta / 2, r_0]}\), a contradiction against that \(r'\) was the infimum.
So, \(\gamma_1 \vert_{(- \infty, r_0]} = \gamma_2 \vert_{(- \infty, r_0]}\).
Step 8:
Let us suppose that \(J\) is lower bounded with the lower boundary, \(r_1\).
Step 9:
Note that \(J\) may be lower open or lower closed.
What we will do is parallel to the case that \(J\) is not lower bounded.
Let us define \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\), where the infimum is taken in \(\mathbb{R}\), which exists as \(r' \in \mathbb{R}\), because \(J\) is lower bounded.
Let us suppose that \(r'\) was not the lower boundary, which would mean \(r_1 \lt r'\).
\(r' \in Int (J)\) whether \(J\) is lower open or lower closed, obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_1, r_0)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' + \delta / 2) \neq \gamma_2 (r' + \delta / 2)\), a contradiction against that \(r'\) was the infimum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).
\(f\) is locally Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r' - \delta / 2, r_0]} = \gamma_2 \vert_{[r' - \delta / 2, r_0]}\), a contradiction against that \(r'\) was the infimum.
So, \(r'\) is the lower boundary, which means \(r' = r_1\).
That means that \(\gamma_1 \vert_{(r_1, r_0]} = \gamma_2 \vert_{(r_1, r_0]}\) or \(\gamma_1 \vert_{[r_1, r_0]} = \gamma_2 \vert_{[r_1, r_0]}\) according to whether \(J\) is lower open or is lower closed: \(\gamma_1 (r_1) = \gamma_2 (r_1)\) can be proved as before.
Step 10:
So, \(\gamma_1 \vert_J = \gamma_2 \vert_J\).
