A definition of locally trivial surjection of rank \(r\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of surjection.
- The reader knows a definition of neighborhood of point.
- The reader knows a definition of \(C^\infty\) manifold.
- The reader knows a definition of diffeomorphism.
- The reader knows a definition of finite-dimensional real vectors space.
- The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a definition of locally trivial surjection of rank \(r\).
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: Definition
For any \(C^\infty\) manifolds, \(E\) and \(M\), and any natural number, \(r\), any surjection, \(\pi: E \to M\), such that 1) for any point, \(p \in M\), the preimage of \(p\), \(\pi^{-1} (p)\), has any structure of \(r\)-dimensional real vectors space and 2) for any \(p \in M\), there are any neighborhood of \(p\), \(U_p\), and any diffeomorphism, \(\phi: \pi^{-1} (U_p) \to U_p \times \mathbb{R}^n\), such that for any \(q \in U_p\), its restriction to \(\pi^{-1} (q)\) is \(\phi|_{\pi^{-1} (q)}: \pi^{-1} (q) \to \{q\} \times \mathbb{R}^r\) and is any 'vectors spaces - linear morphisms' isomorphism
