A description/proof of that for product of 2 \(C^\infty\) manifolds, product for which one of constituents is replaced with regular submanifold is regular submanifold
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 the product of any 2 \(C^\infty\) manifolds, the product for which one of the constituents is replaced with any regular submanifold is a regular submanifold of the original product.
Orientation
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There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any \(C^\infty\) manifolds, \(T'_1, T'_2\), and the product, \(T'_1 \times T'_2\), for any regular submanifold, \(T_1 \subseteq T'_1\), \(T_1 \times T'_2\) is a regular submanifold of \(T'_1 \times T'_2\); for any regular submanifold, \(T_2 \subseteq T'_2\), \(T'_1 \times T_2\) is a regular submanifold of \(T'_1 \times T'_2\).
2: Proof
For any point, \(p = (p_{T_1}, p_{T'_2}) \in T_1 \times T'_2\), there is an adopted chart, \((U'_{p_{T_1}} \subseteq T'_1, \phi'_{p_{T_1}})\) and a chart, \((U'_{p_{T'_2}} \subseteq T'_2, \phi'_{p_{T'_2}})\). Let us take a chart, \((U'_{p_{T_1}} \times U'_{p_{T'_2}} \subseteq T'_1 \times T'_2, \phi'_{p_{T_1}} \times \phi'_{p_{T'_2}})\). \((U'_{p_{T_1}} \times U'_{p_{T'_2}}) \cap (T_1 \times T'_2) = (U'_{p_{T_1}} \cap T_1) \times (U'_{p_{T'_2}} \cap T'_2) = (U'_{p_{T_1}} \cap T_1) \times U'_{p_{T'_2}} = U_{p_{T_1}} \times U'_{p_{T'_2}}\) where \((U_{p_{T_1}} \subseteq T_1, \phi_{p_{T_1}})\) is the adopting chart that corresponds to \(U'_{p_{T_1}}\), by the proposition that the intersection of the same-indices-set products of possibly uncountable number of sets is the product of the intersections of the sets. \(U_{p_{T_1}} \times U'_{p_{T'_2}} = \{p' \in U'_{p_{T_1}} \times U'_{p_{T'_2}}\vert \phi'_{p_{T_1}} \times \phi'_{p_{T'_2}} (p') = (x^1, x^2, . . ., x^{d_1}, 0, 0, . . ., 0, y^1, y^2, . . ., y^{d'_2})\}\) where \(x^i\) and \(y^i\) are coordinates of \(\phi'_{p_{T_1}}\) and \(\phi'_{p_{T'_2}}\) and \(d_1\) and \(d'_2\) are the dimensions of \(T_1\) and \(T'_2\). So, \((U'_{p_{T_1}} \times U'_{p_{T'_2}} \subseteq T'_1 \times T'_2, \phi'_{p_{T_1}} \times \phi'_{p_{T'_2}})\) is an adopted chart and \((U_{p_{T_1}} \times U'_{p_{T'_2}} \subseteq T_1 \times T'_2, \phi_{p_{T_1}} \times \phi'_{p_{T'_2}})\) is the corresponding adopting chart.
It is likewise for \(T'_1 \times T_2\).
