definition of finite-product \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of product topology.
- The reader knows a definition of product map.
Target Context
- The reader will have a definition of finite-product \(C^\infty\) manifold with boundary.
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:
\( \{M_1, ..., M_{n - 1}\}\): \(\subseteq \{\text{ the } C^\infty \text{ manifolds }\}\), with the atlases, \(\{A_1, ..., A_{n - 1}\}\)
\( M_n\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\), with the atlas, \(A_n\)
\(*M_1 \times ... \times M_n\): \(= \text{ the } C^\infty \text{ manifold with boundary }\) with the product topology and the atlas, \(A\), specified later
\(f\): \(: \mathbb{R}^{d_1} \times ... \times \mathbb{R}^{d_n} \to \mathbb{R}^{d_1 + ... + d_n}\), \(= \text{ the canonical homeomorphism }\)
\(g\): \(: \mathbb{R}^{d_1} \times ... \times \mathbb{H}^{d_n} \to \mathbb{H}^{d_1 + ... + d_n}\), \(= \text{ the canonical homeomorphism }\)
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Conditions:
\(\forall (U_{j, \alpha_j} \subseteq M_j, \phi_{j, \alpha_j}) \in A_j ((U_{1, \alpha_1} \times ... \times U_{n, \alpha_n} \subseteq M_1 \times ... \times M_n, (f \text{ or } g) \circ \phi_{1, \alpha_1} \times ... \times \phi_{n, \alpha_n}) \in A)\), where \(f\) or \(g\) depends on whether \((U_{n, \alpha_n} \subseteq M_n, \phi_{j, \alpha_n})\) is an inner chart or a boundary chart
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2: Natural Language Description
For any \(C^\infty\) manifolds, \(M_1, ..., M_{n - 1}\), with any atlases, \(A_1, ..., A_{n - 1}\), any \(C^\infty\) manifold with boundary, \(M_n\), with any atlas, \(A_n\), the canonical homeomorphism, \(f: \mathbb{R}^{d_1} \times ... \times \mathbb{R}^{d_n} \to \mathbb{R}^{d_1 + ... + d_n}\), and the canonical homeomorphism, \(g: \mathbb{R}^{d_1} \times ... \times \mathbb{H}^{d_n} \to \mathbb{H}^{d_1 + ... + d_n}\), the \(C^\infty\) manifold with boundary, \(M_1 \times ... \times M_n\), with the product topology and the atlas, \(A\), such that \(\forall (U_{j, \alpha_j} \subseteq M_j, \phi_{j, \alpha_j}) \in A_j ((U_{1, \alpha_1} \times ... \times U_{n, \alpha_n} \subseteq M_1 \times ... \times M_n, (f \text{ or } g) \circ \phi_{1, \alpha_1} \times ... \times \phi_{n, \alpha_n}) \in A)\), where \(f\) or \(g\) depends on whether \((U_{n, \alpha_n} \subseteq M_n, \phi_{j, \alpha_n})\) is an inner chart or a boundary chart
3: Note
The boundary of \(M_n\) may be empty, then, \(M_n\) will be a \(C^\infty\) manifold, and \(M_1 \times ... \times M_n\) will be a \(C^\infty\) manifold. So, this definition includes 'finite-product \(C^\infty\) manifold'.
Let us see that the definition is well-defined.
The product of any Hausdorff topological spaces is a Hausdorff topological space, by the proposition that the product of any possibly uncountable number of Hausdorff topological spaces is Hausdorff.
The product of any finite number of 2nd-countable topological spaces is a 2nd-countable topological space, by the proposition that the product of any finite number of 2nd-countable topological spaces is 2nd-countable.
\(U_{1, \alpha_1} \times ... \times U_{n, \alpha_n} \subseteq M_1 \times ... \times M_n\) is open, by the definition of product topology.
\(f\) and \(g\) are indeed homeomorphisms, by the proposition that the \(d\)-dimensional Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces whose (the product's) dimension equals \(d\) and the proposition that the \(d\)-dimensional closed upper half Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces and a closed upper half Euclidean space whose (the product's) dimension equals \(d\).
\((f \text{ or } g) \circ \phi_{1, \alpha_1} (U_{1, \alpha_1}) \times ... \times \phi_{n, \alpha_n} (U_{n, \alpha_n}) \subseteq \mathbb{R}^{d_1 + ... + d_n} \text{ or } \mathbb{H}^{d_1 + ... + d_n}\) is open, by the definition of product topology, because \(\phi_{j, \alpha_j} (U_{j, \alpha_j}) \in \mathbb{R}^{d_j} \text{ or } \mathbb{H}^{d_j}\) is open.
\((f \text{ or } g) \circ \phi_{1, \alpha_1} \times ... \times \phi_{n, \alpha_n}\) is a homeomorphism, by the proposition that the product map of any finite number of continuous maps is continuous by the product topologies and the proposition that the preimage by any product map is the product of the preimages by the component maps: as each component map is a homeomorphism, the preimage of each point on the product codomain is the point that is the product of the images under the inverses of the component maps, so, the inverse of the product map is the product of the continuous inverses of the component maps.
As for the transition map, \((f \text{ or } g) \circ \phi'_{1, \alpha_1} \times ... \times \phi'_{n, \alpha_n} \circ ((f \text{ or } g) \circ \phi_{1, \alpha_1} \times ... \times \phi_{n, \alpha_n})^{-1} = (f \text{ or } g) \circ \phi'_{1, \alpha_1} \times ... \times \phi'_{n, \alpha_n} \circ ({\phi_{1, \alpha_1}}^{-1} \times ... \times {\phi_{n, \alpha_n}}^{-1} \circ (f \text{ or } g)^{-1})\), by the proposition that the preimage by any product map is the product of the preimages by the component maps, and \(= (f \text{ or } g) \circ (\phi'_{1, \alpha_1} \circ {\phi_{1, \alpha_1}}^{-1}) \times ... \times (\phi'_{n, \alpha_n} \circ {\phi_{n, \alpha_n}}^{-1}) \circ (f \text{ or } g)^{-1}\), by the proposition that any composition of product maps is the product map of the compositions of the component maps, which is \(C^\infty\) because each \(\phi'_{j, \alpha_j} \circ {\phi_{j, \alpha_j}}^{-1}\) is \(C^\infty\).
The charts cover \(M_1 \times ... \times M_n\), because for each \((m_1, ..., m_n) \in M_1 \times ... \times M_n\), there is a chart, \((U_{j, \alpha_j} \subseteq M_j, \phi_{j, \alpha_j}) \in A_j\), such that \(m_j \in U_{j, \alpha_j}\) for each \(j \in \{1, ..., n\}\), and \((m_1, ..., m_n) \in U_{1, \alpha_1} \times ... \times U_{n, \alpha_n}\).
