description/proof of that for finite-product \(C^\infty\) manifold with boundary, projection is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any finite-product \(C^\infty\) manifold with boundary, each projection is \(C^\infty\).
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:
\(\{M_1, ..., M_{n - 1}\}\): \(\subseteq \{\text{ the } C^\infty \text{ manifolds }\}\)
\(M_n\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_1 \times ... \times M_n\): \(= \text{ the product } C^\infty \text{ manifold with boundary }\)
\(\pi_j\): \(: M_1 \times ... \times M_n \to M_j\), \(= \text{ the projection }\)
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Statements:
\(\pi_j \in \{\text{ the } C^\infty \text{ maps }\}\)
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2: Proof
Whole Strategy: Step 1: for each \(m = (m^1, ..., m^n) \in M_1 \times ... \times M_n\), take a chart, \((U_{1, m^1} \times ... \times U_{n, m^n} \subseteq M_1 \times ... \times M_n, (f \text{ or } g) \circ \phi_{1, m^1} \times ... \times \phi_{n, m^n})\), and the chart, \((U_{j, m^j} \subseteq M_j, \phi_{j, m^j})\); Step 2: see that \(\phi_{j, m^j} \circ \pi_j \circ ((f \text{ or } g) \circ \phi_{1, m^1} \times ... \times \phi_{n, m^n})^{-1}\) is \(C^\infty\).
Step 1:
Let \(m = (m^1, ..., m^n) \in M_1 \times ... \times M_n\) be any.
Let \(f: \mathbb{R}^{d_1} \times ... \times \mathbb{R}^{d_n} \to \mathbb{R}^{d_1 + ... + d_n}\) and \(g: \mathbb{R}^{d_1} \times ... \times \mathbb{R}^{d_{n - 1}} \times \mathbb{H}^{d_n} \to \mathbb{H}^{d_1 + ... + d_n}\) be the canonical homeomorphisms.
There are a chart, \((U_{1, m^1} \times ... \times U_{n, m^n} \subseteq M_1 \times ... \times M_n, (f \text{ or } g) \circ \phi_{1, m^1} \times ... \times \phi_{n, m^n})\), and the chart, \((U_{j, m^j} \subseteq M_j, \phi_{j, m^j})\), by the definition of finite-product \(C^\infty\) manifold with boundary.
Step 2:
\(\pi_j (U_{1, m^1} \times ... \times U_{n, m^n}) \subseteq U_{j, m^j}\).
Let us see that \(\phi_{j, m^j} \circ \pi_j \circ ((f \text{ or } g) \circ \phi_{1, m^1} \times ... \times \phi_{n, m^n})^{-1}: (f \text{ or } g) \circ \phi_{1, m^1} \times ... \times \phi_{n, m^n} (U_{1, m^1} \times ... \times U_{n, m^n}) \subseteq \mathbb{R}^{d_1 + ... + d_n} \text{ or } \mathbb{H}^{d_1 + ... + d_n} \to \phi_{j, m^j} (U_{j, m^j}) \subseteq \mathbb{R}^{d_j} \text{ or } \mathbb{H}^{d_j}\), where when \(j \lt n\), \(\mathbb{R}^{d_j}\), and when \(j = n\), \(\mathbb{H}^{d_j}\), is \(C^\infty\).
It is \((f \text{ or } g) (\phi_{1, m^1} (m'^1), ..., \phi_{n, m^n} (m'^n)) = (\phi_{1, m^1}^1 (m'^1), ..., \phi_{1, m^1}^{d_1} (m'^1), ..., \phi_{n, m^n}^1 (m'^n), ..., \phi_{n, m^n}^{d_n} (m'^n)) \mapsto (\phi_{j, m^j}^1 (m'^j), ..., \phi_{j, m^j}^{d_j} (m'^j))\).
That is obviously \(C^\infty\).
So, \(\pi_j\) is \(C^\infty\).
