description/proof of that \(C^\infty\) immersion is locally \(C^\infty\) embedding
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 \(C^\infty\) embedding.
- The reader knows a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this.
- The reader admits the proposition that for any finite-dimensional columns or rows vectors space and any linearly independent subset, the subset can be shrunk into a number-of-elements-dimensional columns or rows vectors space by choosing some components.
- The reader admits the inverse theorem for Euclidean-normed spaces map.
- The reader admits the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace.
Target Context
- The reader will have a description and a proof of the proposition that any \(C^\infty\) immersion is locally a \(C^\infty\) embedding.
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:
\(M_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ immersions }\}\)
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Statements:
\(\forall m \in M_1 (\exists U_m \in \{\text{ the open neighborhoods of } m \text{ on } M_1\} (f \vert_{U_m}: U_m \to M_2 \in \{\text{ the } C^\infty \text{ embeddings }\}))\)
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2: Proof
Whole Strategy: Step 1: take any chart around \(f (m)\), \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), and any chart around \(m\) (interior when \(m\) is any interior point and boundary when \(m\) is any boundary point), \((U'_m \subseteq M_1, \phi'_m)\), such that \(f (U'_m) \subseteq U_{f (m)}\); Step 2: take \(f': U'_{\phi'_m (m)} \subseteq \mathbb{R}^{d_1} \to \mathbb{R}^{d_2}\) where \(f' := \phi_{f (m)} \circ f \circ {\phi'_m}^{-1}\) when \(m\) is any interior point and \(f' \vert_{U'_{\phi'_m (m)} \cap \phi'_m (U'_m)} = \phi_{f (m)} \circ f \circ {\phi'_m}^{-1} \vert_{U'_{\phi'_m (m)} \cap \phi'_m (U'_m)}\) when \(m\) is any boundary point, take a projection, \(\pi: \mathbb{R}^{d_2} \to \mathbb{R}^{d_1}\), such that \(d (\pi \circ f'): \mathbb{R}^{d_1} \to \mathbb{R}^{d_1}\) is rank \(d_1\); Step 3: apply the the inverse theorem for Euclidean-normed spaces map to get a diffeomorphic \(f'' := \pi \circ f' \vert_{U_{\phi'_m (m)}}: U_{\phi'_m (m)} \to U_{\pi \circ \phi_{f (m)} (f (m))}\); Step 4: take \(U_m := {\phi'_m}^{-1} (U_{\phi'_m (m)} \cap \phi'_m (U'_m))\) and see that \(f''' := f \vert_{U_m}: U_m \to f (U_m)\) is a homeomorphism; Step 5: conclude the proposition.
Step 1:
Let us take any chart around \(f (m)\), \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\).
Let us take any chart around \(m\), \((U'_m \subseteq M_1, \phi'_m)\), such that \(f (U'_m) \subseteq U_{f (m)}\), which is possible, because \(f\) is continuous: when \(m\) is any interior point of \(M_1\), let the chart be an interior char; when \(m\) is any boundary point of \(M_1\), the chart is inevitably a boundary chart.
Step 2:
When \(m\) is any interior point of \(M_1\), let \(U'_{\phi'_m (m)} := \phi'_m (U'_m)\), which is an open neighborhood of \(\phi'_m (m)\) on \(\mathbb{R}^{d_1}\), and \(f': U'_{\phi'_m (m)} \to \mathbb{R}^{d_2} = \phi_{f (m)} \circ f \circ {\phi'_m}^{-1}\).
When \(m\) is any boundary point of \(M_1\), there is a \(C^\infty\) \(f': U'_{\phi'_m (m)} \to \mathbb{R}^{d_2}\) such that \(f' \vert_{U'_{\phi'_m (m)} \cap \phi'_m (U'_m)} = \phi_{f (m)} \circ f \circ {\phi'_m}^{-1} \vert_{U'_{\phi'_m (m)} \cap \phi'_m (U'_m)}\), by the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
Anyway, the differential of \(f'\), \(d f': \mathbb{R}^{d_1} \to \mathbb{R}^{d_2}\), is an injection at \(\phi'_m (m)\): when \(m\) is any boundary point of \(M_1\), the Jacobian of \(f'\), \((\partial_l f'^j)\), which determines the components function of \(d f'\), does not depend on the choice of \(f'\), because \(\partial_{d_1} f'^j\) equals the one-sided \(\partial_{d_1} (\phi_{f (m)} \circ f \circ {\phi'_m}^{-1})\), and the Jacobian is rank \(d_1\).
So, there is a projection, \(\pi: \mathbb{R}^{d_2} \to \mathbb{R}^{d_1}\), such that the differential of \(\pi \circ f': U'_{\phi'_m (m)} \to \mathbb{R}^{d_1}\), \(d (\pi \circ f'): \mathbb{R}^{d_1} \to \mathbb{R}^{d_1}\), is rank \(d_1\) at \(\phi'_m (m)\): \(d f'\)'s having the rank, \(d_1\), means that \(d f' (\mathbb{R}^{d_1})\) is a \(d_1\)-dimensional vectors subspace of \(\mathbb{R}^{d_2}\), so, there is a basis for the subspace, and there is a projection, \(\pi: \mathbb{R}^{d_2} \to \mathbb{R}^{d_1}\), under which the image of the basis is linearly independent, by the proposition that for any finite-dimensional columns or rows vectors space and any linearly independent subset, the subset can be shrunk into a number-of-elements-dimensional columns or rows vectors space by choosing some components (any projection is nothing but choosing some components), and \(\pi \circ d f': \mathbb{R}^{d_1} \to \mathbb{R}^{d_1}\) is rank \(d_1\), because its range is the linear combinations of the projection of the basis, then, \(d (\pi \circ f') = d \pi \circ d f' = \pi \circ d f'\), because \(d \pi = \pi\): while \(\pi: (r^1, ..., r^{d_2})^t \mapsto (r^{j_1}, ..., r^{j_{d_1}})^t = (M^{j_1}_l r^l, ..., M^{j_{d_1}}_l r^l)^t\) where \(M\) is the \(d_1 \times d_2\) matrix such that \(M^m_l\) is \(1\) when the \(l\)-th component of \(\mathbb{R}^{d_2}\) is projected to the \(m\)-th component of \(\mathbb{R}^{d_1}\) and is \(0\) otherwise, \(\partial_n \pi^m = \partial_n M^{j_m}_l r^l = M^{j_m}_n\), so, \(d \pi: (r^1, ..., r^{d_2})^t \mapsto (M^{j_1}_l r^l, ..., M^{j_{d_1}}_l r^l)^t\): refer to the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this.
Step 3:
So, \(\pi \circ f'\) is a \(C^\infty\) map from an open subset of \(\mathbb{R}^{d_1}\), \(U'_{\phi'_m (m)}\), into \(\mathbb{R}^{d_1}\) whose Jacobian is non-singular, so, by the inverse theorem for Euclidean-normed spaces map, there are an open neighborhood of \(\phi'_m (m)\), \(U_{\phi'_m (m)} \subseteq U'_{\phi'_m (m)}\), and an open neighborhood of \(\pi \circ \phi_{f (m)} (f (m))\), \(U_{\pi \circ \phi_{f (m)} (f (m))} \subseteq \mathbb{R}^{d_1}\), such that \(f'' := \pi \circ f' \vert_{U_{\phi'_m (m)}}: U_{\phi'_m (m)} \to U_{\pi \circ \phi_{f (m)} (f (m))}\) is a bijection with the \(C^\infty\) inverse, \(f''^{-1}: U_{\pi \circ \phi_{f (m)} (f (m))} \to U_{\phi'_m (m)}\).
Step 4:
Let us take \(U_m := {\phi'_m}^{-1} (U_{\phi'_m (m)} \cap \phi'_m (U'_m)) \subseteq U'_m\), which is an open neighborhood of \(m\) on \(U'_m\) and on \(M_1\), because \(U_{\phi'_m (m)} \cap \phi'_m (U'_m)\) is an open neighborhood of \(\phi'_m (m)\) on \(\phi'_m (U'_m)\): when \(m\) is any boundary point, \(\phi'_m (U'_m) \subseteq \mathbb{H}^{d_1}\) is a subspace of \(\mathbb{H}^{d_1}\), and \(U_{\phi'_m (m)} \cap \phi'_m (U'_m) = U_{\phi'_m (m)} \cap \phi'_m (U'_m) \cap \mathbb{H}^{d_1} = (U_{\phi'_m (m)} \cap \mathbb{H}^{d_1}) \cap \phi'_m (U'_m)\), where \(U_{\phi'_m (m)} \cap \mathbb{H}^{d_1}\) is an open subset of \(\mathbb{H}^{d_1}\).
Let us take \(f''' := f \vert_{U_m}: U_m \to f (U_m)\).
\(f'''\) is injective, because otherwise, \(f''\) would not be injective: for each \(p \in U_m\), \(\phi'_m (p) \in \phi'_m (U'_m)\), so, \(f'' \vert_{\phi'_m (U_m)} = \pi \circ f' \vert_{\phi'_m (U_m)} = \pi \circ \phi_{f (m)} \circ f \circ {\phi'_m}^{-1} \vert_{\phi'_m (U_m)} = \pi \circ \phi_{f (m)} \circ f \vert_{U_m} \circ {\phi'_m}^{-1} \vert_{\phi'_m (U_m)} = \pi \circ \phi_{f (m)} \circ f''' \circ {\phi'_m}^{-1} \vert_{\phi'_m (U_m)}\).
So, \(f'''\) is bijective.
\(f'''^{-1}: f (U_m) \to U_m = {\phi'_m}^{-1} \circ f''^{-1} \circ \pi \circ \phi_{f (m)} \vert_{f (U_m)}\), which is valid and true, because for each \(f (p) \in f (U_m)\), \(\pi \circ \phi_{f (m)} (f (p)) = \pi \circ \phi_{f (m)} \circ f \circ {\phi'_m}^{-1} \circ \phi'_m (p)\), but as \(\phi'_m (p) \in U_{\phi'_m (m)} \cap \phi'_m (U'_m) \subseteq U'_{\phi'_m (m)} \cap \phi'_m (U'_m)\), \(= \pi \circ f' \circ \phi'_m (p) = f'' \circ \phi'_m (p)\), and \({\phi'_m}^{-1} \circ f''^{-1} \circ \pi \circ \phi_{f (m)} \vert_{f (U_m)} = {\phi'_m}^{-1} \circ f''^{-1} \circ f'' \circ \phi'_m (p) = {\phi'_m}^{-1} \circ \phi'_m (p) = p\).
\(f'''\) is continuous as a restriction of a continuous map.
\(f'''^{-1}\) is continuous as a composition of continuous maps, because \(\pi \circ \phi_{f (m)} \vert_{f (U_m)}: f (U_m) \subseteq M_2 \to U_{\pi \circ \phi_{f (m)} (f (m))} \subseteq \mathbb{R}^{d_1}\) is continuous as a restriction of continuous \(\pi \circ \phi_{f (m)}: U_{f (m)} \subseteq M_2 \to \phi_{f (m)} (U_{f (m)}) \subseteq \mathbb{R}^{d_2} \to \mathbb{R}^{d_1}\), \(f''^{-1}: U_{\pi \circ \phi_{f (m)} (f (m))} \subseteq \mathbb{R}^{d_1} \to U_{\phi'_m (m)} \subseteq \mathbb{R}^{d_1}\) is continuous because it is \(C^\infty\) and \(f''^{-1}: U_{\pi \circ \phi_{f (m)} (f (m))} \subseteq \mathbb{R}^{d_1} \to \phi'_m (U_m) \subseteq \mathbb{R}^{d_1} \text{ or } \mathbb{H}^{d_1}\) is continuous as the codomain restriction (\(\phi'_m (U_m) \subseteq \mathbb{R}^{d_1} \text{ or } \mathbb{H}^{d_1}\) is a topological subspace of \(U_{\phi'_m (m)} \subseteq \mathbb{R}^{d_1}\), because as \(\mathbb{H}^{d_1}\) is a topological subspace of \(\mathbb{R}^{d_1}\), \(\phi'_m (U_m)\) is a topological subspace of \(\mathbb{R}^{d_1}\) anyway, by the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace, and the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace again applies for \(\phi'_m (U_m) \subseteq U_{\phi'_m (m)} \subseteq \mathbb{R}^{d_1}\)), and \({\phi'_m}^{-1} \vert_{\phi'_m (U_m)}: \phi'_m (U_m) \subseteq \mathbb{R}^{d_1} \text{ or } \mathbb{H}^{d_1} \to U_m\) is continuous.
So, \(f'''\) is a homeomorphism.
Step 5:
\(f \vert_{U_m}: U_m \to M_2\) is a \(C^\infty\) immersion where \(f''' := f \vert_{U_m}: U_m \to f (U_m)\) is a homeomorphism, so, \(f \vert_{U_m}\) is a \(C^\infty\) embedding.
