description/proof of that for \(d'\)-dimensional \(C^\infty\) manifold with boundary, \(d\)-dimensional immersed submanifold with boundary, and point on submanifold, if there is chart on base manifold whose restriction on submanifold is constant w.r.t \(d' - d\) components, tangent space on submanifold is spanned by \(d\) components of standard basis
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of immersed submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows standard basis for tangent vectors space at point on \(C^\infty\) manifold with boundary with respect to chart.
- 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 the range of any linear map between any vectors spaces is a vectors subspace of the codomain.
- The reader admits the proposition that any bijective linear map between any vectors spaces is a 'vectors spaces - linear morphisms' isomorphism.
- The reader admits the proposition that for any 'vectors spaces - linear morphisms' isomorphism, the image of any linearly independent subset or any basis of the domain is linearly independent or a basis on the codomain.
- The reader admits the proposition that for any vectors space over any field and any square matrix over the field with dimension equal to or smaller than the dimension of the vectors space, the matrix is invertible if it maps a linearly-independent set of vectors to a linearly-independent set of vectors, and if the matrix is invertible, it maps any linearly-independent set of vectors to a linearly-independent set of vectors.
Target Context
- The reader will have a description and a proof of the proposition that for any \(d'\)-dimensional \(C^\infty\) manifold with boundary, any \(d\)-dimensional immersed submanifold with boundary, and any point on the submanifold, if there is any chart around the point on the base manifold whose restriction on the submanifold domain is constant with respect to any \(d' - d\) components, the image of the tangent space at the point on the submanifold under the inclusion differential is spanned by the \(d\) components of the standard basis.
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:
\(M'\): \(\in \{\text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(M\): \(\in \{\text{ the } d \text{ -dimensional immersed submanifolds with boundary of } M'\}\)
\(\iota\): \(: M \to M'\), \(= \text{ the inclusion }\)
\(m\): \(\in M\)
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Statements:
\(\exists (U'_m \subseteq M', \phi'_m) \in \{\text{ the charts around } m \text{ for } M'\} (\phi'_m \vert_{U'_m \cap M}: U'_m \cap M \to \phi'_m (U'_m), m' \mapsto (y^1, ..., y^d, c^{d + 1}, ..., c^{d'}))\), where \(\{c^{d + 1}, ..., c^{d'}\}\) are some constants
\(\implies\)
\(d \iota T_mM = Span (\partial / \partial y^1, ..., \partial / \partial y^d)\)
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2: Proof
Whole Strategy: Step 1: take any chart around \(m\) on \(M\), \((U_m \subseteq M, \phi_m)\), such that \(\iota (U_m) \subseteq U'_m\), and see that \(d \iota \vert_{T_mM}: v^j \partial / \partial x^j \mapsto \partial \hat{\iota}^j / \partial x^l v^l \partial / \partial y^j\) where \(\hat{\iota} = \phi'_m \circ \iota \circ {\phi_m}^{-1}\); Step 2: see that \(d \iota T_mM = span (\partial / \partial y^1, ..., \partial / \partial y^d)\).
Step 1:
Let us take any chart around \(m\) on \(M\), \((U_m \subseteq M, \phi_m)\), such that \(\iota (U_m) \subseteq U'_m\), which is possible because \(\iota\) is continuous.
\(d \iota \vert_{T_mM}: v^j \partial / \partial x^j \mapsto \partial \hat{\iota}^j / \partial x^l v^l \partial / \partial y^j\) where \(\hat{\iota} = \phi'_m \circ \iota \circ {\phi_m}^{-1}\), by 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 2:
By the supposition, for each \(j \in \{d + 1, ..., d'\}\) and each \(l \in \{1, ..., d\}\), \(\partial \hat{\iota}^j / \partial x^l = 0\).
So, \(\partial \hat{\iota}^j / \partial x^l v^l \partial / \partial y^j = \sum_{j \in \{1, ..., d\}} \partial \hat{\iota}^j / \partial x^l v^l \partial / \partial y^j\).
So, \(d \iota T_mM \subseteq Span (\{\partial / \partial y^1, ..., \partial / \partial y^d\})\).
\(d \iota T_mM\) is a vectors subspace of the vectors space spanned by \(\{\partial / \partial y^1, ..., \partial / \partial y^d\}\), by the proposition that the range of any linear map between any vectors spaces is a vectors subspace of the codomain.
As \(d \iota \vert_{T_mM}\) is a linear injection, \(d \iota \vert_{T_mM}: T_mM \to d \iota \vert_{T_mM} (T_mM)\) is a linear bijection, and is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that any bijective linear map between any vectors spaces is a 'vectors spaces - linear morphisms' isomorphism, and \(\{d \iota \vert_{T_mM} \partial / \partial x^1, ..., d \iota \vert_{T_mM} \partial / \partial x^d\}\) is a basis for \(d \iota T_mM\), by the proposition that for any 'vectors spaces - linear morphisms' isomorphism, the image of any linearly independent subset or any basis of the domain is linearly independent or a basis on the codomain.
As each element of \(d \iota T_mM\) is expressed as \(\sum_{j \in \{1, ..., d\}} \partial \hat{\iota}^j / \partial x^l v^l \partial / \partial y^j\), there is a \(d \times d\) real matrix, \(M\), such that \(\begin{pmatrix} d \iota \vert_{T_mM} \partial / \partial x^1 \\ ... \\ d \iota \vert_{T_mM} \partial / \partial x^d \end{pmatrix} = M \begin{pmatrix} \partial / \partial y^1 \\ ... \\ \partial / \partial y^d \end{pmatrix}\).
As \((d \iota \vert_{T_mM} \partial / \partial x^1, ..., d \iota \vert_{T_mM} \partial / \partial x^d)\) is linearly-independent and \((\partial / \partial y^1, ..., \partial / \partial y^d)\) is linearly-independent, \(M\) is invertible, by the proposition that for any vectors space over any field and any square matrix over the field with dimension equal to or smaller than the dimension of the vectors space, the matrix is invertible if it maps a linearly-independent set of vectors to a linearly-independent set of vectors, and if the matrix is invertible, it maps any linearly-independent set of vectors to a linearly-independent set of vectors.
So, \(\begin{pmatrix} \partial / \partial y^1 \\ ... \\ \partial / \partial y^d \end{pmatrix} = M^{-1} \begin{pmatrix} d \iota \vert_{T_mM} \partial / \partial x^1 \\ ... \\ d \iota \vert_{T_mM} \partial / \partial x^d \end{pmatrix}\).
So, \(\{\partial / \partial y^1, ..., \partial / \partial y^d\} \subseteq d \iota T_mM\), and so, \(Span (\{\partial / \partial y^1, ..., \partial / \partial y^d\}) \subseteq d \iota T_mM\), because \(d \iota T_mM\) is a vectors space.
So, \(d \iota T_mM = Span (\partial / \partial y^1, ..., \partial / \partial y^d)\).
