description/proof of that for \(C^\infty\) immersion between \(C^\infty\) manifolds with boundary, its global differential is \(C^\infty\) immersion
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) immersion.
- The reader knows a definition of global differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary.
- The reader knows a definition of double of \(C^\infty\) manifold with boundary.
- The reader admits the rank theorem for \(C^\infty\) immersion.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) immersion between any \(C^\infty\) manifolds with boundary, its global differential is a \(C^\infty\) immersion.
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:
\(M_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ immersions }\}\)
\(d f\): \(: T M_1 \to T M_2\), \(= \text{ the global differential }\)
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Statements:
\(d f \in \{\text{ the } C^\infty \text{ immersions }\}\).
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2: Proof
Whole Strategy: do it in the 2 parts: the 1st part: \(M_2\) has the empty boundary; the 2nd part: \(M_2\) has a nonempty boundary; Step 1: suppose that \(M_2\) has the empty boundary; Step 2: around each \(m \in M_1\), take a chart, \((U_m \subseteq M_1, \phi_m)\), and a chart, \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), such that \(\widehat{f} := \phi_{f (m)} \circ f \circ {\phi_m}^{-1}\) is \(: (x^1, ..., x^{d_1}) \mapsto (x^1, ..., x^{d_1}, 0, ..., 0)\); Step 3: let \(\pi_1: T M_1 \to M_1\) and \(\pi_2: T M_2 \to M_2\) be the projections; take the induced charts, \(({\pi_1}^{-1} (U_m) \subseteq T M_1, \widetilde{\phi_m})\) and \(({\pi_2}^{-1} (U_{f (m)}) \subseteq T M_2, \widetilde{\phi_{f (m)}})\), and see that \(\widehat{d f} := \widetilde{\phi_{f (m)}} \circ d f \circ {\widetilde{\phi_m}}^{-1}\) is \(: (v^1, ..., v^k, x^1, ..., x^{d_1}) \mapsto (v^1, ..., v^k, 0, ..., 0, x^1, ..., x^{d_1}, 0, ..., 0)\); Step 4: see that \(d f\) is a \(C^\infty\) immersion; Step 5: suppose that \(M_2\) has a nonempty boundary; Step 6: take the double of \(M_2\), \(D (M_2)\), let \(\widetilde{M_2} \subseteq D (M_2)\) be a regular domain diffeomorphic to \(M_2\), with a diffeomorphism, \(g: M_2 \to \widetilde{M_2}\), let \(\widetilde{\iota}: \widetilde{M_2} \to D (M_2)\) be the inclusion, take \(h := \widetilde{\iota} \circ g \circ f\), apply the 1st part conclusion to see that \(d h\) is a \(C^\infty\) immersion, and see that \(d f\) is a \(C^\infty\) immersion.
Step 1:
Le us suppose that \(M_2\) has the empty boundary.
Step 2:
Around each \(m \in M_1\), let us take a chart, \((U_m \subseteq M_1, \phi_m)\), and a chart, \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), such that \(f (U_m) \subseteq U_{f (m)}\) and \(\widehat{f} := \phi_{f (m)} \circ f \circ {\phi_m}^{-1}: \phi_m (U_m) \to \phi_{f (m)} (U_{f (m)})\), the coordinates function of \(f\), is \(: (x^1, ..., x^{d_1}) \mapsto (x^1, ..., x^{d_1}, 0, ..., 0)\), which is possible by the rank theorem for \(C^\infty\) immersion (which requires \(M_2\) to be without boundary, which is the reason why we have made the supposition of Step 1).
Step 3:
Let \(\pi_1: T M_1 \to M_1\) and \(\pi_2: T M_2 \to M_2\) be the projections.
Let us take the induced charts, \(({\pi_1}^{-1} (U_m) \subseteq T M_1, \widetilde{\phi_m})\) and \(({\pi_2}^{-1} (U_{f (m)}) \subseteq T M_2, \widetilde{\phi_{f (m)}})\).
It is a well-known fact that \(\widehat{d f} := \widetilde{\phi_{f (m)}} \circ d f \circ {\widetilde{\phi_m}}^{-1}: \widetilde{\phi_m} ({\pi_1}^{-1} (U_m)) \to \widetilde{\phi_{f (m)}} ({\pi_2}^{-1} (U_{f (m)}))\) is \(: (v^1, ..., v^{d_1}, x^1, ..., x^{d_1}) \mapsto (v^1, ..., v^{d_1}, 0, ..., 0, x^1, ..., x^{d_1}, 0, ..., 0)\): the components, \((v^1, ..., v^{d_1})\), are mapped to \((\partial_j \widehat{f}^1 v^j, ..., \partial_j \widehat{f}^{d_2} v^j)\), but \(\widehat{f}^j = x^j\) for each \(1 \le j \le d_1\) and \(\widehat{f}^j = 0\) for each \(d_1 + 1 \le j \le d_2\).
So, \(d f\) is \(C^\infty\).
Step 4:
Let \(\pi'_1: T T M_1 \to T M_1\) and \(\pi'_2: T T M_2 \to T M_2\) be the projections.
There are the induced charts, \(({\pi'_1}^{-1} ({\pi_1}^{-1} (U_m)) \subseteq T T M_1, \widetilde{\widetilde{\phi_m}})\) and \(({\pi'_2}^{-1} ({\pi_2}^{-1} (U_{f (m)})) \subseteq T T M_2, \widetilde{\widetilde{\phi_{f (m)}}})\).
It is a well-known fact that \(\widehat{d d f} := \widetilde{\widetilde{\phi_{f (m)}}} \circ d d f \circ {\widetilde{\widetilde{\phi_m}}}^{-1}: \widetilde{\widetilde{\phi_m}} ({\pi'_1}^{-1} ({\pi_1}^{-1} (U_m))) \to \widetilde{\widetilde{\phi_{f (m)}}} ({\pi'_2}^{-1} ({\pi_2}^{-1} (U_{f (m)})))\) is \(: (w^1, ..., w^{2 d_1}, v^1, ..., v^{d_1}, x^1, ..., x^{d_1}) \mapsto (w^1, ..., w^{d_1}, 0, ..., 0, w^{d_1 + 1}, ..., w^{2 d_1}, 0, ..., 0, v^1, ..., v^{d_1}, 0, ..., 0, x^1, ..., x^{d_1}, 0, ..., 0)\): the components, \((w^1, ..., w^{2 d_1})\), are mapped to \((\partial_j \widehat{d f}^1 w^j, ..., \partial_j \widehat{d f}^{2 d_2} w^j)\) where \(\partial_j\) s for \(1 \le j \le d_1\) are by \(v^j\) s and \(\partial_j\) s for \(d_1 + 1 \le j \le 2 d_1\) are by \(x^{j - d_1}\) s, but \(\widehat{d f}^j = v^j\) for each \(1 \le j \le d_1\), \(\widehat{d f}^j = 0\) for each \(d_1 + 1 \le j \le d_2\), \(\widehat{d f}^j = x^{j - d_2}\) for each \(d_2 + 1 \le j \le d_2 + d_1\), and \(\widehat{d f}^j = 0\) for each \(d_2 + d_1 + 1 \le j \le 2 d_2\).
That is obviously injective.
So, \(d f\) is a \(C^\infty\) immersion.
Step 5:
Let us suppose that \(M_2\) has a nonempty boundary.
Step 6:
Let us take the double of \(M_2\), \(D (M_2)\).
\(D (M_2)\) is a \(C^\infty\) manifold without boundary that has a regular domain, \(\widetilde{M_2}\), that is diffeomorphic to \(M_2\).
Let a diffeomorphism be \(g: M_2 \to \widetilde{M_2}\).
Let \(\widetilde{\iota}: \widetilde{M_2} \to D (M_2)\) be the inclusion, which is a \(C^\infty\) immersion (in fact, a \(C^\infty\) embedding), because \(\widetilde{M_2}\) is an embedded submanifold (in fact, a regular domain) of \(D (M_2)\).
Let us think of \(h: M_1 \to D (M_2) = \widetilde{\iota} \circ g \circ f: M_1 \to M_2 \to \widetilde{M_2} \to D (M_2)\).
\(h\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
\(h\) is a \(C^\infty\) immersion, because \(d h = d \widetilde{\iota} \circ d g \circ d f\) and \(d f\), \(d g\), and \(d \widetilde{\iota}\) are injective.
By the 1st part conclusion, \(d h = d \widetilde{\iota} \circ d g \circ d f\) is a \(C^\infty\) immersion, which means that \(d d h = d d \widetilde{\iota} \circ d d g \circ d d f\) is injective on each fiber.
Then, \(d d f\) is injective on each fiber, because otherwise, the 2 vectors that were mapped to the same vector under \(d d f\) would not be mapped to any different vectors under \(d d h\).
So, \(d f\) is a \(C^\infty\) immersion.
