description/proof of that for \(C^\infty\) manifold with boundary, tangent vectors bundle, and immersed submanifold with boundary of base space, tangent vectors bundle of submanifold with boundary is canonically \(C^\infty\) embedded into restricted vectors bundle
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vectors bundle over \(C^\infty\) manifold with boundary.
- The reader knows a definition of immersed submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows a definition of restricted \(C^\infty\) vectors bundle.
- The reader knows a definition of \(C^\infty\) embedding.
- The reader admits the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary, the global differential is \(C^\infty\).
- The reader admits the proposition that for any \(C^\infty\) immersion between any \(C^\infty\) manifolds with boundary, its global differential is a \(C^\infty\) immersion.
- The reader admits the proposition that for any n x n matrix, if there are any m rows with more than any n - m same columns 0, the matrix is not invertible.
- The reader admits the proposition that for any invertible square matrix, from the top row downward through any row, each row can be changed to have a 1 1 component and 0 others without any duplication to keep the matrix invertible.
- The reader admits the proposition that in order to check the continuousness of any map between any topological spaces, the preimages of only any basis or any subbasis are enough to be checked.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that for any topological space and its 2 products with any Euclidean topological spaces, any injective continuous map between the products fiber-preserving and linear on each fiber is a continuous embedding.
- The reader admits the proposition that any expansion of any continuous embedding on the codomain is a continuous embedding.
- The reader admits the proposition that any injective map between any topological spaces is a continuous embedding if the domain restriction of the map on each element of any open cover is any continuous embedding onto any open subset of the range or the codomain.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, its tangent vectors bundle, and any immersed submanifold with boundary of the base space, the tangent vectors bundle of the submanifold with boundary is canonically \(C^\infty\) embedded into the restricted vectors bundle.
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:
\(M\): \(\in \{\text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\((T M, M, \pi')\): \(= \text{ the tangent vectors bundle of } M\)
\(S\): \(\in \{\text{ the } d \text{ -dimensional immersed submanifolds with boundary of } M\}\)
\(\iota\): \(: S \to M\), \(= \text{ the inclusion }\)
\(T M \vert_S\): \(= \text{ the restricted } C^\infty \text{ vectors bundle }\), with \(\pi: T M \vert_S \to S\) as the projection
\((T S, S, \rho)\): \(= \text{ the tangent vectors bundle of } S\)
\(\iota'\): \(: T S \to T M \vert_S, v \mapsto d \iota v\)
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Statements:
\(\iota' \in \{\text{ the } C^\infty \text{ embeddings }\}\)
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2: Note
Note that although \(T M \vert_S\) is an immersed submanifold with boundary of \(T M\) (refer to Note for the definition of restricted \(C^\infty\) vectors bundle), it is not any embedded submanifold with boundary of \(T M\) in general, which is why the proof of this proposition is not so simple.
3: Proof
Whole Strategy: Step 1: see that \(\iota'\) is injective; Step 2: according to the definition of restricted \(C^\infty\) vectors bundle, for each \(s \in S\), take a chart around \(s\), \((U_\beta \subseteq S, \phi_\beta)\), the induced chart, \((\rho^{-1} (U_\beta) \subseteq T S, \widetilde{\phi_\beta})\), and a chart, \((\pi^{-1} (U_\beta) \subseteq T M \vert_S, \overline{\phi_\beta})\), and see what the components function of \(\iota'\) is like; Step 3: see that \(\iota'\) is \(C^\infty\); Step 4: see what \(d \iota'\) is like the components-wise, and see that \(d \iota'\) is injective on each fiber; Step 5: see that the codomain restriction of \(\iota'\), \(\iota'': T S \to \iota' (T S) \subseteq T M \vert_S\) is homeomorphic; Step 6: conclude the proposition.
Step 1:
Let us see that \(\iota'\) is injective.
As \(\iota\) is a \(C^\infty\) immersion, \(d \iota: T S \to T M\) is injective.
Although the codomain of \(\iota'\) is \(T M \vert_S\) instead of \(T M\), the injectiveness is only about the map between-sets-wise, and the injectiveness of \(\iota'\) holds.
Step 2:
Let us take a chart for \(T S\) and a chart for \(T M \vert_S\) in order to study the local behaviors of \(\iota'\) via the coordinates function.
Let \(s \in S\) be any.
According to the definition of restricted \(C^\infty\) vectors bundle, we can take a chart, \((U_\beta \subseteq S, \phi_\beta)\), and a chart, \((\pi^{-1} (U_\beta) \subseteq T M \vert_S, \overline{\phi_\beta})\).
\(\overline{\phi_\beta}\) is induced by a trivialization for \(T M\), \(\Phi'_\beta: \pi'^{-1} (U'_\beta) \to U'_\beta \times \mathbb{R}^{d'}\), where \(U'_\beta \subseteq M\) is a trivializing open subset. In fact, we can take \(U'_\beta\) as the domain of a chart, \((U'_\beta \subseteq M, \phi'_\beta)\), and we do so, and we can take the trivialization as the canonical one induced by the chart, and we do so.
Let us take the canonical chart induced by \((U_\beta \subseteq S, \phi_\beta)\), \((\rho^{-1} (U_\beta) \subseteq T S, \widetilde{\phi_\beta})\).
Let us take the canonical chart induced by \((U'_\beta \subseteq M, \phi'_\beta)\), \((\pi'^{-1} (U'_\beta) \subseteq T M, \widetilde{\phi'_\beta})\).
\(\iota' (\rho^{-1} (U_\beta)) \subseteq \pi^{-1} (U_\beta)\), because \(\iota'\) is fiber-preserving, because \(d \iota\) is so.
So, we are going to study the components of \(\iota'\) with respect to \((\rho^{-1} (U_\beta) \subseteq T S, \widetilde{\phi_\beta})\) and \((\pi^{-1} (U_\beta) \subseteq T M \vert_S, \overline{\phi_\beta})\).
In fact, the components function is of the form, \(: (v^1, ..., v^d, x^1, ..., x^d) \mapsto (w^1, ..., w^{d'}, x^1, ..., x^d)\), because the both charts use the same \(\phi_\beta\).
Step 3:
As a comparison, let us think of the global differential of \(\iota\), \(d \iota: T S \to T M\).
The components function of \(d \iota\) with respect to \((\rho^{-1} (U_\beta) \subseteq T S, \widetilde{\phi_\beta})\) and \((\pi'^{-1} (U'_\beta) \subseteq T M, \widetilde{\phi'_\beta})\), is of the form, \(: (v^1, ..., v^d, x^1, ..., x^d) \mapsto (w^1, ..., w^{d'}, y^1, ..., y^{d'})\), whose point is that \((w^1, ..., w^{d'})\) is exactly the same with the components function of \(\iota'\), which is because while both \(\iota'\) and \(d \iota\) do the mappings with the same \(d \iota\), both \((\pi^{-1} (U_\beta) \subseteq T M \vert_S, \overline{\phi_\beta})\) and \((\pi'^{-1} (U'_\beta) \subseteq T M, \widetilde{\phi'_\beta})\) are induced by the same trivialization, \(\Phi'_\beta\).
\(d \iota\) is \(C^\infty\), by the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary, the global differential is \(C^\infty\).
That means that \(w^j\) is a \(C^\infty\) function of \((v^1, ..., v^d, x^1, ..., x^d)\).
Then, also \(: (v^1, ..., v^d, x^1, ..., x^d) \mapsto (w^1, ..., w^{d'}, x^1, ..., x^d)\) is \(C^\infty\).
So, \(\iota'\) is \(C^\infty\).
Step 4:
There are the canonical chart for \(T T S\) induced by \((\rho^{-1} (U_\beta) \subseteq T S, \widetilde{\phi_\beta})\) and the canonical chart for \(T T M \vert_S\) induced by \((\pi^{-1} (U_\beta) \subseteq T M \vert_S, \overline{\phi_\beta})\).
As is well-known, the components function of \(d \iota'\) with respect to the charts is of the form, \(: (t^1, ..., t^{2 d}, x^1, ..., x^d, v^1, ..., v^d) \mapsto (\partial_j x^1 t^j, ..., \partial_j x^d t^j, \partial_j w^1 t^j, ..., \partial_j w^{d'} t^j, x^1, ..., x^d, w^1, ..., w^{d'})\), where \(\partial_j\) is by \(x^j\) for \(j \in \{1, ..., d\}\) and by \(v^{j - d}\) for \(j \in \{d + 1, ..., 2 d\}\), \(= (t^1, ..., t^d, \partial_j w^1 t^j, ..., \partial_j w^{d'} t^j, x^1, ..., x^d, w^1, ..., w^{d'})\).
As a comparison, let us think of \(d d \iota\).
There is the canonical chart for \(T T M\) induced by \((\pi'^{-1} (U'_\beta) \subseteq T M, \widetilde{\phi'_\beta})\).
The components function of \(d d \iota\) with respect to the chart for \(T T S\) and the chart for \(T T M\) is of the form, \(: (t^1, ..., t^{2 d}, x^1, ..., x^d, v^1, ..., v^d) \mapsto (\partial_j y^1 t^j, ..., \partial_j y^{d'} t^j, \partial_j w^1 t^j, ..., \partial_j w^{d'} t^j, y^1, ..., y^{d'}, w^1, ..., w^{d'})\), where \(\partial_j\) is by \(x^j\) for \(j \in \{1, ..., d\}\) and by \(v^{j - d}\) for \(j \in \{d + 1, ..., 2 d\}\), but as \(y\) depends only on \(x\), \(\partial_j y^l t^j\) s are only for \(j \in \{1, ..., d\}\).
\(d d \iota\) is injective on each fiber, because \(d \iota\) is a \(C^\infty\) immersion, by the proposition that for any \(C^\infty\) immersion between any \(C^\infty\) manifolds with boundary, its global differential is a \(C^\infty\) immersion.
That means that the matrix, \(\begin{pmatrix} \partial y^1 / \partial x^1 & ... & \partial y^1 / \partial x^d & 0 & ... & 0 \\ ... \\ \partial y^{d'} / \partial x^1 & ... & \partial y^{d'} / \partial x^d & 0 & ... & 0 \\ \partial w^1 / \partial x^1 & ... & \partial w^1 / \partial x^d & \partial w^1 / \partial v^1 & ... & \partial w^1 / \partial v^d \\ ... \\ \partial w^{d'} / \partial x^1 & ... & \partial w^{d'} / \partial x^d & \partial w^{d'} / \partial v^1 & ... & \partial w^{d'} / \partial v^d \end{pmatrix}\), is rank \(2 d\).
That means that there is a \(2 d \times 2 d\) submatrix, \(\begin{pmatrix} \partial y^{j_1} / \partial x^1 & ... & \partial y^{j_1} / \partial x^d & 0 & ... & 0 \\ ... \\ \partial y^{j_k} / \partial x^1 & ... & \partial y^{j_k} / \partial x^d & 0 & ... & 0 \\ \partial w^{l_1} / \partial x^1 & ... & \partial w^{l_1} / \partial x^d & \partial w^{l_1} / \partial v^1 & ... & \partial w^{l_1} / \partial v^d \\ ... \\ \partial w^{l_m} / \partial x^1 & ... & \partial w^{l_m} / \partial x^d & \partial w^{l_m} / \partial v^1 & ... & \partial w^{l_m} / \partial v^d \end{pmatrix}\), that is invertible.
In fact, \(k \le d\), because otherwise, the submatrix would not be invertible by the proposition that for any n x n matrix, if there are any m rows with more than any n - m same columns 0, the matrix is not invertible: if \(d \lt k\), \(2 d - k \lt d\), but as the \(k\) rows had the \(d\) same columns 0, the submatrix would not be invertible.
Each of the top \(k\) rows can be changed to have a 1 1 component and 0 others without any duplication to keep the matrix invertible, by the proposition that for any invertible square matrix, from the top row downward through any row, each row can be changed to have a 1 1 component and 0 others without any duplication to keep the matrix invertible.
That is a \(2 d \times 2 d\) submatrix of the matrix for the components function of \(d \iota'\), which means that the matrix is rank \(2 d\), which implies that \(d \iota'\) is injective on each fiber.
So, \(\iota'\) is a \(C^\infty\) immersion.
Step 5:
\(\{\rho^{-1} (U_\beta) \vert \beta \in B\}\) is an open cover of \(T S\).
Let \(\Phi_\beta: \pi^{-1} (U_\beta) \to U_\beta \times \mathbb{R}^{d'}\) be the trivialization for \(T M \vert_S\) defined in the definition of restricted \(C^\infty\) vectors bundle.
Let \(\Phi''_\beta: \rho^{-1} (U_\beta) \to U_\beta \times \mathbb{R}^d\) be the canonical trivialization for \(T S\).
Let us think of \(\Phi_\beta \circ \iota' \vert_{\rho^{-1} (U_\beta)} \circ {\Phi''_\beta}^{-1}: U_\beta \times \mathbb{R}^d \to U_\beta \times \mathbb{R}^{d'}\).
It is an injective continuous map fiber-preserving and linear on each fiber, and so, is a continuous embedding, by the proposition that for any topological space and its 2 products with any Euclidean topological spaces, any injective continuous map between the products fiber-preserving and linear on each fiber is a continuous embedding.
So, \(\iota' \vert_{\rho^{-1} (U_\beta)} = {\Phi_\beta}^{-1} \circ \Phi_\beta \circ \iota' \vert_{\rho^{-1} (U_\beta)} \circ {\Phi''_\beta}^{-1} \circ \Phi''_\beta: \rho^{-1} (U_\beta) \to \pi^{-1} (U_\beta)\) is a continuous embedding: \(\Phi''_\beta\) and \(\Phi_\beta\) are some homeomorphisms. Also the codomain expansion, \(\iota' \vert_{\rho^{-1} (U_\beta)}: \rho^{-1} (U_\beta) \to T M \vert_S\), is a continuous embedding, by the proposition that any expansion of any continuous embedding on the codomain is a continuous embedding.
\(\iota' \vert_{\rho^{-1} (U_\beta)} (\rho^{-1} (U_\beta)) = \pi^{-1} (U_\beta) \cap \iota' (T S)\), which is open on \(T M \vert_S \cap \iota' (T S)\), because \(\pi^{-1} (U_\beta)\) is open on \(T M \vert_S\).
So, \(\iota'\) is a continuous embedding, by the proposition that any injective map between any topological spaces is a continuous embedding if the domain restriction of the map on each element of any open cover is any continuous embedding onto any open subset of the range or the codomain.
That means that the codomain restriction of \(\iota'\), \(\iota'': T S \to \iota' (T S) \subseteq T M \vert_S\), is a homeomorphism.
Step 6:
So, \(\iota'\) is an injective immersion whose codomain restriction is a homeomorphism, which means that \(\iota'\) is a \(C^\infty\) embedding.
