363: For Product of 2 C^\infty Manifolds, Product for Which One of Constituents Is Replaced with Regular Submanifold Is Regular Submanifold

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A description/proof of that for product of 2 $C^{\mathrm{\infty }}$ manifolds, product for which one of constituents is replaced with regular submanifold is regular submanifold

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of product $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of regular submanifold.
• The reader admits 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.

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Target Context

• The reader will have a description and a proof of the proposition that for the product of any 2 $C^{\mathrm{\infty }}$ manifolds, the product for which one of the constituents is replaced with any regular submanifold is a regular submanifold of the original product.

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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.

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Main Body

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1: Description

For any $C^{\mathrm{\infty }}$ manifolds, $T_1^{\prime },T_2^{\prime }$, and the product, $T_1^{\prime }\times T_2^{\prime }$, for any regular submanifold, $T_1\subseteq T_1^{\prime }$, $T_1\times T_2^{\prime }$ is a regular submanifold of $T_1^{\prime }\times T_2^{\prime }$; for any regular submanifold, $T_2\subseteq T_2^{\prime }$, $T_1^{\prime }\times T_2$ is a regular submanifold of $T_1^{\prime }\times T_2^{\prime }$.

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2: Proof

For any point, $p=(p_{T_1},p_{T_2^{\prime }})\in T_1\times T_2^{\prime }$, there is an adopted chart, $(U_{p_{T_1}}^{\prime }\subseteq T_1^{\prime },{\varphi }_{p_{T_1}}^{\prime })$ and a chart, $(U_{p_{T_2^{\prime }}}^{\prime }\subseteq T_2^{\prime },{\varphi }_{p_{T_2^{\prime }}}^{\prime })$. Let us take a chart, $(U_{p_{T_1}}^{\prime }\times U_{p_{T_2^{\prime }}}^{\prime }\subseteq T_1^{\prime }\times T_2^{\prime },{\varphi }_{p_{T_1}}^{\prime }\times {\varphi }_{p_{T_2^{\prime }}}^{\prime })$. $(U_{p_{T_1}}^{\prime }\times U_{p_{T_2^{\prime }}}^{\prime })\cap (T_1\times T_2^{\prime })=(U_{p_{T_1}}^{\prime }\cap T_1)\times (U_{p_{T_2^{\prime }}}^{\prime }\cap T_2^{\prime })=(U_{p_{T_1}}^{\prime }\cap T_1)\times U_{p_{T_2^{\prime }}}^{\prime }=U_{p_{T_1}}\times U_{p_{T_2^{\prime }}}^{\prime }$ where $(U_{p_{T_1}}\subseteq T_1,{\varphi }_{p_{T_1}})$ is the adopting chart that corresponds to $U_{p_{T_1}}^{\prime }$, 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. $U_{p_{T_1}}\times U_{p_{T_2^{\prime }}}^{\prime }=\{p^{\prime }\in U_{p_{T_1}}^{\prime }\times U_{p_{T_2^{\prime }}}^{\prime }|{\varphi }_{p_{T_1}}^{\prime }\times {\varphi }_{p_{T_2^{\prime }}}^{\prime }(p^{\prime })=(x^1,x^2,...,x^{d_1},0,0,...,0,y^1,y^2,...,y^{d_2^{\prime }})\}$ where $x^i$ and $y^i$ are coordinates of ${\varphi }_{p_{T_1}}^{\prime }$ and ${\varphi }_{p_{T_2^{\prime }}}^{\prime }$ and $d_1$ and $d_2^{\prime }$ are the dimensions of $T_1$ and $T_2^{\prime }$. So, $(U_{p_{T_1}}^{\prime }\times U_{p_{T_2^{\prime }}}^{\prime }\subseteq T_1^{\prime }\times T_2^{\prime },{\varphi }_{p_{T_1}}^{\prime }\times {\varphi }_{p_{T_2^{\prime }}}^{\prime })$ is an adopted chart and $(U_{p_{T_1}}\times U_{p_{T_2^{\prime }}}^{\prime }\subseteq T_1\times T_2^{\prime },{\varphi }_{p_{T_1}}\times {\varphi }_{p_{T_2^{\prime }}}^{\prime })$ is the corresponding adopting chart.

It is likewise for $T_1^{\prime }\times T_2$.

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References

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