definition of orientation-preserving local diffeomorphism between oriented \(C^\infty\) manifolds with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of orientation-preserving local diffeomorphism between oriented \(C^\infty\) manifolds with boundary.
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:
\( (M_1, o_1)\): \(\in \{\text{ the } d \text{ -dimensional oriented } C^\infty \text{ manifolds with boundary }\}\)
\( (M_2, o_2)\): \(\in \{\text{ the } d \text{ -dimensional oriented } C^\infty \text{ manifolds with boundary }\}\)
\(*f\): \(: M_1 \to M_2\), \(\in \{\text{ the local diffeomorphisms }\}\)
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Conditions:
\(\forall m_1 \in M_1 (\forall (v_1, ..., v_d) \in \{\text{ the oriented bases for } T_{m_1}M_1\} ((d f v_1, ..., d f v_d) \in \{\text{ the oriented bases for } T_{f (m_1)}M_2\}))\)
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2: Note
\((d f v_1, ..., d f v_d)\) is inevitably a basis as far as \(f\) is a local diffeomorphism, 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.
While the definition makes the requirement for each oriented \((v_1, ..., v_d)\), if the requirement is satisfied for an oriented \((v'_1, ..., v'_d)\), the requirement is inevitably satisfied for each oriented \((v_1, ..., v_d)\), because \(v_j = M^l_j v'_l\) for a matrix, \(M\), such that \(0 \lt det M\), and \(d f v_j = d f (M^l_j v'_l) = M^l_j d f (v'_l)\), because \(d f\) is linear, and so, as \((d f v'_1, ..., d f v'_d)\) is oriented, \((d f v_1, ..., d f v_d)\) is oriented.
