definition of \(J\)-half-slice of chart domain w.r.t. point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(J\)-slice of chart domain with respect to point.
Target Context
- The reader will have a definition of \(J\)-half-slice of chart domain with respect to point.
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:
\( M\): \(\in \{\text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\( (U \subseteq M, \phi)\): \(\in \{\text{ the charts for } M\}\)
\( J\): \(\subseteq \{1, ..., d'\}\), \(= (j_1, ..., j_d)\)
\( u\): \(\in U\) such that \(\phi (u)^{j_d} = 0\)
\( H_{J, \phi (u)} (\mathbb{R}^{d'})\): \(= \{r \in \mathbb{R}^{d'} \vert \forall j \in \{1, ..., d'\} \setminus J (r^j = \phi (u)^j) \land 0 \le r^{j_d}\}\)
\(*H_{J, u} (U)\): \(\subseteq U\)
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Conditions:
\(\phi (H_{J, u} (U)) = \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})\)
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2: Note
\(H_{J, u} (U)\) inevitably exists as a nonempty subset for any \(J\) and any \(u\) (as far as \(\phi (u)^{j_d} = 0\) is satisfied, which requires that the chart is chosen to allow such a \(u\)), because \(\phi (u) \in \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})\) and \(H_{J, u} (U) = \phi^{-1} (\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}))\).
\(H_{J, \phi (u)} (\mathbb{R}^{d'})\) is like \(\{(\phi (u)^1, ..., x^{j_1}, ..., x^{j_d}, ..., \phi (u)^{d'}) \vert \forall j_l \in J \setminus \{j_d\} (x^{j_l} \in \mathbb{R}), x^{j_d} \in \mathbb{H}\}\), where whether the 1st component is really \(\phi (u)^1\) or \(x^{j_1}\) and whether the last component is really \(\phi (u)^{d'}\) or \(x^{j_d}\) depend on \(J\).
Let \(\pi_J: \mathbb{R}^{d'} \to \mathbb{R}^d, (x^1, ..., x^{j_1}, ..., x^{j_d}, ..., x^{d'}) \mapsto (x^{j_1}, ..., x^{j_d})\) be the projection, where whether the 1st component is really \(x^1\) or \(x^{j_1}\) and whether the last component is really \(x^{d'}\) or \(x^{j_d}\) depend on \(J\).
\(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'})}: H_{J, \phi (u)} (\mathbb{R}^{d'}) \subseteq \mathbb{R}^{d'} \to \mathbb{H}^d\) is a diffeomorphism, because it is bijective and \(C^\infty\) (there are the standard chart for \(\mathbb{R}^{d'}\) and the standard chart for \(\mathbb{H}^d\) such that the components function is \(C^\infty\), because the components function has the \(C^\infty\) extension, \(\pi_J: \mathbb{R}^{d'} \to \mathbb{R}^d\)), and the inverse is \(C^\infty\) (there are the standard chart for \(\mathbb{H}^d\) and the standard chart for \(\mathbb{R}^{d'}\) such that the components function is \(C^\infty\), because the components function has the \(C^\infty\) extension, the inverse of \(\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})}: \mathbb{R}^{d'} \to \mathbb{R}^d\)). Especially, \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'})}\) is a homeomorphism.
When \(d' \in J\), \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}: H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'} \to \mathbb{H}^d\) is a diffeomorphism: it is like \(: (\phi (u)^1, ..., x^{j_1}, ..., x^{j_d}) \mapsto (x^{j_1}, ..., x^{j_d})\) where \(0 \le x^{j_d}\), which is bijective and \(C^\infty\) (there are the standard chart for \(\mathbb{H}^{d'}\) and the standard chart for \(\mathbb{H}^d\) such that the components function is \(C^\infty\), because the components function has the \(C^\infty\) extension, \(\pi_J: \mathbb{R}^{d'} \to \mathbb{R}^d\)), and the inverse is \(C^\infty\) (there are the standard chart for \(\mathbb{H}^d\) and the standard chart for \(\mathbb{H}^{d'}\) such that the components function is \(C^\infty\), because the components function has the \(C^\infty\) extension, the inverse of \(\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})}: \mathbb{R}^{d'} \to \mathbb{R}^d\)). Especially, \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}\) is a homeomorphism.
When \(d' \notin J\) and \(0 \le \phi (u)^{d'}\) (we do not need the \(\phi (u)^{d'} \lt 0\) case, because we need this only for when the chart is a boundary chart, which guaranteed that \(0 \le \phi (u)^{d'}\)), \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}: H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'} \to \mathbb{H}^d\) is a diffeomorphism: it is like \(: (\phi (u)^1, ..., x^{j_1}, ..., x^{j_d}, ..., \phi (u)^{d'}) \mapsto (x^{j_1}, ..., x^{j_d})\) where \(0 \le x^{j_d}\), which is bijective and \(C^\infty\) (there are the standard chart for \(\mathbb{H}^{d'}\) and the standard chart for \(\mathbb{H}^d\) such that the components function is \(C^\infty\), because the components function has the \(C^\infty\) extension, \(\pi_J: \mathbb{R}^{d'} \to \mathbb{R}^d\)), and the inverse is \(C^\infty\) (there are the standard chart for \(\mathbb{H}^d\) and the standard chart for \(\mathbb{H}^{d'}\) such that the components function is \(C^\infty\), because the components function has the \(C^\infty\) extension, the inverse of \(\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})}: \mathbb{R}^{d'} \to \mathbb{R}^d\)). Especially, \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}\) is a homeomorphism. The condition, \(0 \le \phi (u)^{d'}\), is necessary, because otherwise, \(H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} = \emptyset\), which would not be homeomorphic to \(\mathbb{H}^d\).
Let us suppose that the chart is an interior chart.
\(\phi (U)\) is an open subset of \(\mathbb{R}^{d'}\).
So, \(\phi (H_{J, u} (U))\) is an open subset of \(H_{J, \phi (u)} (\mathbb{R}^{d'}) \subseteq \mathbb{R}^{d'}\).
So, \(\pi_J (\phi (H_{J, u} (U))) = \pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'})} (\phi (H_{J, u} (U)))\) is an open subset of \(\mathbb{H}^d\).
\(\pi_J \circ \phi \vert_{H_{J, u} (U)}: H_{J, u} (U) \subseteq U \to \pi_J (\phi (H_{J, u} (U))) \subseteq \mathbb{H}^d\) is a homeomorphism, because it is \(\pi_J \vert_{\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})} \circ \phi \vert_{H_{J, u} (U)}\), and \(\phi \vert_{H_{J, u} (U)}: H_{J, u} (U) \subseteq U \to \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \subseteq \phi (U)\) and \(\pi_J \vert_{\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})}: \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \subseteq H_{J, \phi (u)} (\mathbb{R}^{d'}) \to \pi_J (\phi (H_{J, u} (U))) \subseteq \mathbb{H}^d\) are some homeomorphisms as some restrictions of homeomorphic \(\phi\) and \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'})}\): \(\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})\) as the codomain of \(\phi \vert_{H_{J, u} (U)}\) is the subspace of \(\phi (U)\), but as \(\phi (U)\) is the subspace of \(\mathbb{R}^{d'}\), the codomain is the subspace of \(\mathbb{R}^{d'}\), 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, while \(\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})\) as the domain of \(\pi_J \vert_{\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'})}\) is the subspace of \(H_{J, \phi (u)} (\mathbb{R}^{d'})\), but as \(H_{J, \phi (u)} (\mathbb{R}^{d'})\) is the subspace of \(\mathbb{R}^{d'}\), the domain is the subspace of \(\mathbb{R}^{d'}\), likewise.
Let us suppose that the chart is a boundary chart.
\(\phi (U)\) is an open subset of \(\mathbb{H}^{d'}\) and \(\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) = \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}\).
So, \(\phi (H_{J, u} (U))\) is an open subset of \(H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'}\).
So, \(\pi_J (\phi (H_{J, u} (U))) = \pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} (\phi (H_{J, u} (U)))\) is an open subset of \(\mathbb{H}^d\).
\(\pi_J \circ \phi \vert_{H_{J, u} (U)}: H_{J, u} (U) \subseteq U \to \pi_J (\phi (H_{J, u} (U))) \subseteq \mathbb{H}^d\) is a homeomorphism, because it is \(\pi_J \vert_{\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ \phi \vert_{H_{J, u} (U)}\), and \(\phi \vert_{H_{J, u} (U)}: H_{J, u} (U) \subseteq U \to \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \phi (U)\) and \(\pi_J \vert_{\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}: \phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \to \pi_J (\phi (H_{J, u} (U))) \subseteq \mathbb{H}^d\) are some homeomorphisms as some restrictions of homeomorphic \(\phi\) and \(\pi_J \vert_{H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}\): \(\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}\) as the codomain of \(\phi \vert_{H_{J, u} (U)}\) is the subspace of \(\phi (U)\), but as \(\phi (U)\) is the subspace of \(\mathbb{H}^{d'}\), the codomain is the subspace of \(\mathbb{H}^{d'}\), as before, while \(\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}\) as the domain of \(\pi_J \vert_{\phi (U) \cap H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}\) is the subspace of \(H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}\), but as \(H_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}\) is the subspace of \(\mathbb{H}^{d'}\), the domain is the subspace of \(\mathbb{H}^{d'}\), likewise.
The purpose of requiring \(\phi (u)^{j_d} = 0\) is only that \(H_{J, u} (U)\) has at least \(1\) boundary point.
It is not that \(H_{J, u} (U)\) would be critically bad if it had no boundary point, but it would be useless for our purpose, because we are considering \(H_{J, u} (U)\) in order to allow a boundary point.
But is that requirement not too restrictive? No, we are considering '\(J\)-half-slice of chart domain with respect to point' in order to have some boundary points (otherwise, '\(J\)-slice of chart domain with respect to point' would be enough), and we can just choose \(u\) as one of them and if \(\phi (u)^{j_d} \neq 0\), we can just translate \(\phi\) to make \(\phi (u)^{j_d} = 0\).
The reason why we have elaborated on those facts is that we are going to construct a chart, \((H_{J, u} (U) \subseteq S, \pi_J \circ \phi \vert_{H_{J, u} (U)})\), for any \(S \subseteq M\) that satisfies a certain condition (called "local-slice-or-half-slice condition"), to make \(S\) an embedded submanifold with boundary of \(M\).
