description/proof of that local diffeomorphism between oriented \(C^\infty\) manifolds with boundary is locally orientation-preserving or orientation-reversing
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of oriented \(C^\infty\) manifold with boundary.
- The reader knows a definition of orientation-preserving local diffeomorphism between oriented \(C^\infty\) manifolds with boundary.
- The reader knows a definition of orientation-reversing local diffeomorphism between oriented \(C^\infty\) manifolds with boundary.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, any section over any trivializing open subset is \(C^\infty\) if and only if the coefficients of the section with respect to any \(C^\infty\) frame over the trivializing open subset are \(C^\infty\).
- 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 diffeomorphism from any \(C^\infty\) manifold with boundary onto any neighborhood of any point image on any \(C^\infty\) manifold with boundary, the differential of the diffeomorphism or any its codomain extension at the point is a 'vectors spaces - linear morphisms' isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that any local diffeomorphism between any oriented \(C^\infty\) manifolds with boundary is locally orientation-preserving or orientation-reversing.
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:
\(M_1\): \(\in \{\text{ the } d \text{ -dimensional oriented } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } d \text{ -dimensional oriented } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the local diffeomorphisms }\}\)
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Statements:
\(\forall m_1 \in M_1 (\exists U_{m_1} \in \{\text{ the open neighborhoods of } m_1\} (f \vert_{U_{m_1}}: U_{m_1} \to f (U_{m_1}) \in \{\text{ the orientation-preserving diffeomorphisms }\} \cup \{\text{ the orientation-reversing diffeomorphisms }\}))\)
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2: Note
Each of the definition of orientation-preserving local diffeomorphism and the definition of orientation-reversing local diffeomorphism has required that the map is orientation-preserving or orientation-reversing at each point of the domain.
Then, is each local diffeomorphism orientation-preserving or orientation-reversing?
No, because the domain may not be connected, and the map may be orientation-preserving over a connected-component and be orientation-reversing over another connected-component.
This proposition is claiming that the map is at least locally orientation-preserving or orientation-reversing.
3: Proof
Whole Strategy: Step 1: take any oriented local \(C^\infty\) frame, \((v_1, ..., v_d)\), over an open neighborhood of \(m_1 \in M_1\), \(U'_{m_1}\), and take a connected chart around \(m_1\), \((U_{m_1} \subseteq M_1, \phi_{m_1})\), such that \(U_{m_1} \subseteq U'_{m_1}\), and see that the chart is 'positively or negatively'-oriented, and likewise, take any oriented local \(C^\infty\) frame around \(f (m_1)\), \((w_1, ..., w_d)\), and take a 'positively or negatively'-oriented connected chart around \(f (m_1)\); Step 2: express \((d f v_1, ..., d f v_d)\) with \((w_1, ..., w_d)\), and see that the determinant of the transition matrix is definitely positive or definitely negative.
Step 1:
There is an oriented local \(C^\infty\) frame, \((v_1, ..., v_d)\), over an open neighborhood of \(m_1\), \(U'_{m_1} \subseteq M_1\).
There is a connected chart around \(m_1\), \((U_{m_1} \subseteq M_1, \phi_{m_1})\), such that \(U_{m_1} \subseteq U'_{m_1}\).
\((\partial / \partial x^1, ..., \partial / \partial x^d)\) is a \(C^\infty\) frame over the trivializing open subset, \(U_{m_1}\).
\(v_j = \partial / \partial x^l M^l_j\) where \(M^l_j\) is \(C^\infty\), by the proposition that for any \(C^\infty\) vectors bundle, any section over any trivializing open subset is \(C^\infty\) if and only if the coefficients of the section with respect to any \(C^\infty\) frame over the trivializing open subset are \(C^\infty\).
\(det M \neq 0\) all over \(U_{m_1}\), because otherwise, \((v_1, ..., v_d)\) would not be linearly independent, so, \(0 \lt det M\) all over \(U_{m_1}\) or \(det M \lt 0\) all over \(U_{m_1}\), because while \(det M: U_{m_1} \to \mathbb{R}\) is \(C^\infty\) (so, continuous), \(det M (U_{m_1})\) is connected.
When \(0 \lt det M\), the chart is positively oriented; when \(det M \lt 0\), the chart is negatively oriented.
Let us do likewise for \(f (m_1) \in M_2\) as follows.
There is an oriented local \(C^\infty\) frame, \((w_1, ..., w_d)\), over an open neighborhood of \(f (m_1)\), \(U'_{f (m_1)} \subseteq M_2\).
There is a connected chart around \(f (m_1)\), \((U_{f (m_1)} \subseteq M_2, \phi_{f (m_1)})\), such that \(U_{f (m_1)} \subseteq U'_{f (m_1)}\).
\((\partial / \partial y^1, ..., \partial / \partial y^d)\) is a \(C^\infty\) frame over the trivializing open subset, \(U_{f (m_1)}\).
\(w_j = \partial / \partial y^l N^l_j\) where \(N^l_j\) is \(C^\infty\), as before.
\(0 \lt det N\) all over \(U_{f (m_1)}\) or \(det N \lt 0\) all over \(U_{f (m_1)}\), as before.
When \(0 \lt det N\), the chart is positively oriented; when \(det N \lt 0\), the chart is negatively oriented.
As \(f\) is a local diffeomorphism, \(U_{m_1}\) and \(U_{f (m_1)}\) can be taken smaller if necessary such that \(f (U_{m_1}) = U_{f (m_1)}\) and \(f \vert_{U_{m_1}}: U_{m_1} \to U_{f (m_1)}\) is a diffeomorphism: take some \(U''_{m_1} \subseteq M_1\) and \(U''_{f (m_1)} \subseteq M_2\) such that \(f \vert_{U''_{m_1}}: U''_{m_1} \to U''_{f (m_1)}\) is a diffeomorphism; take an open neighborhood of \(m\), \(U'''_{m_1} \subseteq M_1\), such that \(f (U'''_{m_1}) \subseteq U_{f (m_1)}\); if \(U''_{m_1} \cap U'''_{m_1} \cap U_{m_1}\) is connected, let it be new \(U_{m_1}\), otherwise, let any connected open neighborhood of \(m_1\) contained in it be new \(U_{m_1}\); let \(f (U_{m_1})\) be new \(U_{f (m_1)}\): the new \(U_{m_1}\) is an open subset of \(U''_{m_1}\) and of \(M_1\), and the new \(U_{f (m_1)}\) is an open subset of \(U''_{f (m_1)}\), so, open on \(M_2\), and the new \(f \vert_{U_{m_1}}: U_{m_1} \to U_{f (m_1)}\) is the open domain restriction of \(f \vert_{U''_{m_1}}: U''_{m_1} \to U''_{f (m_1)}\), so, is a diffeomorphism.
Step 2:
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, \(d f v_n = \partial_l \hat{f}^j M^l_n \partial / \partial y^j\): \(M^l_n\) s are the components of \(v_n\).
As \(w_j = \partial / \partial y^l N^l_j\), \(\partial / \partial y^j = w_s {N^{-1}}^s_j\), where \(N^{-1}\) is the inverse matrix of \(N\).
\(det N^{-1} = 1 / det N\), which is positive or negative according to \(det N\) positive or negative.
\(d f v_n = \partial_l \hat{f}^j M^l_n {N^{-1}}^s_j w_s\).
Let us denote \(\partial_l \hat{f}^j\) as \(O^j_l\).
Note that \(O^j_l\) is really \(: \phi_{m_1} (U_{m_1}) \to \mathbb{R}\), but we will sometimes sloppily state like it is \(: U_{m_1} \to \mathbb{R}\), hereafter: in fact, \(O^j_l \circ \phi_{m_1}\) is so, but such more exact expressions would be probably more disturbing.
\(det O^j_l \neq 0\) all over \(U_{m_1}\), because while \(O\) for each \(m'_1 \in U_{m_1}\) is the components transition matrix for \(d f_{m'_1}: T_{m'_1}M_1 \to T_{f (m'_1)}M_2\), \(d f_{m'_1}\) is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that for any diffeomorphism from any \(C^\infty\) manifold with boundary onto any neighborhood of any point image on any \(C^\infty\) manifold with boundary, the differential of the diffeomorphism or any its codomain extension at the point is a 'vectors spaces - linear morphisms' isomorphism, and the linearly independent \((v_1, ..., v_d)\) is mapped to a linearly independent \((d f v_1, ..., d f v_d)\).
As before, as \(det O: U_{m_1} \to \mathbb{R}\) is \(C^\infty\) and \(U_{m_1}\) is connected, \(0 \lt det O\) all over \(U_{m_1}\) or \(det O \lt 0\) all over \(U_{m_1}\).
\(d f v_n = w_s {N^{-1}}^s_j \partial_l \hat{f}^j M^l_n = w_s {N^{-1}}^s_j O^j_l M^l_n = w_s (N^{-1} O M)^s_n\).
So, the transition matrix from \((w_1, ..., w_d)\) to \((d f v_1, ..., d f v_d)\) is \(N^{-1} O M\), and \(det (N^{-1} O M) = det N^{-1} det O det M\), which is positive all over \(U_{f (m_1)}\) or negative all over \(U_{f (m_1)}\).
So, \(f \vert_{U_{m_1}}\) is orientation-preserving or orientation-reversing.
