description/proof of that for \(C^\infty\) map between \(C^\infty\) manifolds, if map has constant rank \(0\), map is constant over connected component
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of rank of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
- The reader knows a definition of connected topological component.
- The reader admits the rank theorem for \(C^\infty\) map between \(C^\infty\) manifolds.
- The reader admits the proposition that any connected topological component is exactly any connected topological subspace that cannot be made larger.
- The reader admits the proposition that any map that is anywhere locally constant on any connected topological space is globally constant.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds, if the map has the constant rank \(0\), the map is constant over each connected component.
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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
//
Statements:
\(\forall m \in M_1 (Rank (f)_m = 0)\)
\(\implies\)
\(\forall T \in \{\text{ the connected components of } M_1\} (f \vert_T \in \{\text{ the constant maps }\})\)
//
2: Note
\(M_1\) and \(M_2\) are required to be without boundary for this proposition, because this proposition uses the rank theorem for \(C^\infty\) map between \(C^\infty\) manifolds, which requires the domain and the codomain without boundary.
As an immediate corollary, when \(M_1\) is connected, \(f\) is constant.
3: Proof
Whole Strategy: Step 1: see that around each \(m \in M_1\), there is an open neighborhood, \(U_m\), over which \(f\) is constant; Step 2: conclude the proposition.
Step 1:
Let \(m \in M_1\) be any.
As \(f\) has the constant rank, \(0\), by the rank theorem for \(C^\infty\) map between \(C^\infty\) manifolds, there are a chart around \(m\), \((U_m \subseteq M_1, \phi_m)\), and a chart around \(f (m)\), \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), such that \(\phi_{f (m)} \circ f \circ {\phi_m}^{-1} \vert_{\phi_m (U_m)}: \phi_m (U_m) \to \phi_{f (m)} (U_{f (m)})\) is \((x^1, ..., x^{d_1}) \mapsto (0, ..., 0)\).
That means that \(f\) is constant over \(U_m\).
Step 2:
Let \(T \subseteq M_1\) be any connected component of \(M_1\).
\(T\) is a connected topological subspace of \(M_1\), by the proposition that any connected topological component is exactly any connected topological subspace that cannot be made larger.
\(f \vert_T: T \to M_2\) is a map that is anywhere locally constant over a connected topological space: around each \(t \in T\), there is an open neighborhood of \(t\) on \(M_1\), \(U_t \subseteq M_1\), over which \(f\) is constant by Step 1, and \(U_t \cap T \subseteq T\) is an open neighborhood of \(t\) on \(T\) over which \(f \vert_T\) is constant.
By the proposition that any map that is anywhere locally constant on any connected topological space is globally constant, \(f \vert_T\) is globally constant.
