description/proof of that tangent vector at point on \(C^\infty\) manifold with boundary is velocity of \(C^\infty\) curve, especially from half closed interval, especially as linear in coordinates
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
- 4: Proof
Starting Context
- The reader knows a definition of tangent vector.
- The reader knows a definition of velocity of \(C^\infty\) curve at point on \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and any chart, the standard basis for the tangent vectors space at each point on the chart domain is indeed a basis.
Target Context
- The reader will have a description and a proof of the proposition that any tangent vector at any point on any \(C^\infty\) manifold with boundary is the velocity of a \(C^\infty\) curve, especially from a half closed interval, especially as linear in coordinates.
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:
\(M\): \(\in \{ \text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\(m\): \(\in M\)
\(v\): \(\in T_mM\)
\(\mathbb{R}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
//
Statements:
\(\exists J = [t_0, t_2) \text{ or } (t_1, t_0] \subseteq \mathbb{R}, \exists \gamma: J \to M \in \{\text{ the } C^\infty \text{ curves on } M\} (v = d \gamma (d / d t \vert_{t_0}))\)
\(\land\)
Especially, \(\gamma\) can be take to be such that \(t \mapsto {\phi_m}^{-1} ((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)))\), where \((U_m \subseteq M, \phi_m)\) is any chart around \(m\) and \(v^j\) s are the components of \(v\) with respect to the standard basis for the chart
//
2: Natural Language Description
For any \(d\)-dimensional \(C^\infty\) manifold with boundary, \(M\), any point, \(m \in M\), and any tangent vector at \(m\), \(v \in T_mM\), there are an interval, \(J = [t_0, t_2) \text{ or } (t_1, t_0] \subseteq \mathbb{R}\), and a \(C^\infty\) curve, \(\gamma: J \to M\), such that \(v = d \gamma (d / d t \vert_{t_0})\), and especially, \(\gamma\) can be take to be such that \(t \mapsto {\phi_m}^{-1} ((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)))\), where \((U_m \subseteq M, \phi_m)\) is any chart around \(m\) and \(v^j\) s are the components of \(v\) with respect to the standard basis for the chart.
3: Note
A purpose of choosing a non-open interval is to be applicable for all the cases: when \(m\) is a boundary point, we may not be able to choose any curve from open interval.
Of course, there are some cases for which \(J\) can be an open interval, if you want.
Another purpose is to narrow the area only where the values of \(f \in C^\infty (M)\) matter for \(v f\): as we can take \(\gamma\) as \(t \mapsto {\phi_m}^{-1} ((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)))\) while \(0 \le v^d (t - t_0)\), \(v f\) depends on the values of \(f\) only on the upper half of the chart domain (even when \(m\) is not any boundary point), so to speak.
4: Proof
Whole Strategy: Step 1: choose any chart around \(m\), \((U_m \subseteq M, \phi_m)\); Step 2: express \(v\) as \(v^j \partial / \partial x^j\); Step 3: choose a \(J = [t_0, t_2) \text{ or } (t_1, t_0]\) according to \(0 \le v^j\) or \(v^j \lt 0\) and \(\gamma\) as \(t \mapsto {\phi_m}^{-1} ((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)))\); Step 4: see that \(v = d \gamma (d / d t \vert_{t_0})\); Step 5: make some observations.
Step 1:
Let us choose any chart around \(m\), \((U_m \subseteq M, \phi_m)\).
Let \(x = (x^1, ..., x^d)\) be the coordinates on \(\phi_m (U_m)\).
Step 2:
\(v = v^j \partial / \partial x^j\), by the proposition that for any \(C^\infty\) manifold with boundary and any chart, the standard basis for the tangent vectors space at each point on the chart domain is indeed a basis.
Step 3:
Let us choose \(t_0 \in \mathbb{R}\) arbitrarily.
When \(0 \le v^d\), let us choose any \(t_2\) such that \(t_0 \lt t_2\) and \((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)) \in \phi_m (U_m)\) for each \(t \in [t_0, t_2)\), which is possible because there is an open ball, \(B_{\phi_m (m), \epsilon} \subseteq \mathbb{R} \text{ or } \mathbb{H}\), such that \(B_{\phi_m (m), \epsilon} \subseteq \phi_m (U_m)\) and by taking a small enough \(t_2 - t_0\), \((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)) \in B_{\phi_m (m), \epsilon}\): \(0 \le v^d (t - t_0)\), so, even when \(m\) is a boundary point, it holds.
When \(v^d \lt 0\), let us choose any \(t_1\) such that \(t_1 \lt t_0\) and \((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)) \in \phi_m (U_m)\) for each \(t \in (t_1, t_0]\), which is possible because there is an open ball, \(B_{\phi_m (m), \epsilon} \subseteq \mathbb{R} \text{ or } \mathbb{H}\), such that \(B_{\phi_m (m), \epsilon} \subseteq \phi_m (U_m)\) and by taking a small enough \(t_0 - t_1\), \((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)) \in B_{\phi_m (m), \epsilon}\): \(0 \le v^d (t - t_0)\), so, even when \(m\) is a boundary point, it holds.
Let us define \(\gamma: J \to M, t \mapsto {\phi_m}^{-1} ((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)))\).
\(\gamma\) is well-defined because \((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)) \subseteq \phi_m (U_m)\).
\(\gamma\) is \(C^\infty\), because \(J \to \phi_m (U_m), t \mapsto (\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0))\) is obviously \(C^\infty\) and \({\phi_m}^{-1}: \phi_m (U_m) \to M\) is \(C^\infty\).
Step 4:
Let us see that \(v = d \gamma (d / d t \vert_{t_0})\).
Let \(f \in C^\infty (M)\) be any.
\(d \gamma (d / d t \vert_{t_0}) (f) = d / d t \vert_{t_0} (f \circ \gamma) = d / d t \vert_{t_0} (f \circ {\phi_m}^{-1} ((\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0)))) = \partial_j (f \circ {\phi_m}^{-1}) \vert_{\phi_m (m)} d (\phi_m (m)^j + v^j (t - t_0))/ d t \vert_{t_0}\), by the chain rule (in fact, \(\partial_d\) and \(d / d t \vert_{t_0}\) are one-sided), \(= v^j \partial_j (f \circ {\phi_m}^{-1}) \vert_{\phi_m (m)}\).
\(v f = v^j \partial / \partial x^j (f) = v^j \partial_j (f \circ {\phi_m}^{-1}) \vert_{\phi_m (m)}\): see what any chart induced basis vector on any \(C^\infty\) manifold with boundary is.
So, \(v = d \gamma (d / d t \vert_{t_0})\).
Step 5:
Let us make some observations.
\(\phi_m \circ \gamma: J \to \phi_m (U_m)\) is \(t \mapsto (\phi_m (m)^1 + v^1 (t - t_0), ..., \phi_m (m)^d + v^d (t - t_0))\), which is what is meant by \(\gamma\)'s being "linear in coordinates".
As there can be some multiple curves with the same velocity at \(m\), we can take a curve that is not linear in coordinates.
