definition of tangent vectors bundle over \(C^\infty\) 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 tangent vector.
Target Context
- The reader will have a definition of tangent vectors bundle over \(C^\infty\) 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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\( TM\): \(= \cup_{p \in M} T_pM\), \(\in \{\text{ the } 2 d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\), with the topology and the \(C^\infty\) atlas specified below
\( \pi\): \(: TM \to M, v \mapsto p\), where \(v \in T_pM\)
\(*(TM, M, \pi)\):
//
Conditions:
\(\forall (U_\alpha \subseteq M, \phi_\alpha) \in \{\text{ the charts of } M\}\)
(
\(\exists (\pi^{-1} (U_\alpha) \subseteq TM, \phi'_\alpha) \in \{\text{ the charts of } TM\}\)
(
\(\phi'_\alpha: \pi^{-1} (U_\alpha) \to \phi (U_\alpha) \times \mathbb{R}^d \subseteq \mathbb{R}^{2 d} \text{ or } \mathbb{H}^{2 d}, v \mapsto (\phi_\alpha (\pi (v)), \phi''_\alpha (v))\), where \(\phi''_\alpha: \pi^{-1} (U_\alpha) \to \mathbb{R}^d, v \mapsto (v^1, ..., v^d )\), where \(v = v^j \partial x^j\), where \((x^1, ..., x^n)\) are the coordinates of the chart, \((U_\alpha, \phi_\alpha)\)
\(B_\alpha := \{{\phi'_\alpha}^{-1} (U'_\alpha) \vert \forall U'_\alpha \in \{\text{ the open subsets of } \phi_\alpha (U_\alpha) \times \mathbb{R}^d\}\}\)
)
)
\(\land\)
\(B := \cup_{\alpha} B_\alpha \in \{\text{ the topological bases of } TM\}\)
//
As \(\phi'_\alpha\) is obviously bijective, \({\phi'_\alpha}^{-1}\) is valid.
Usually, it is called "tangent bundle", but the author prefers "tangent vectors bundle", because the author prizes expressing explicitly: it is the bundle of the tangent vectors, while "tangent vectors bundle" will be immediately and successfully guessed what it means by anyone who knows what "tangent bundle" is.
2: Natural Language Description
For any \(d\)-dimensional \(C^\infty\) manifold with boundary, \(M\), \((TM, M, \pi)\), where \(TM\) is the \(2 d\)-dimensional \(C^\infty\) manifold with boundary, \(\cup_{p \in M} T_pM\), with the topology and the \(C^\infty\) atlas specified below and \(\pi: TM \to M\) is \(v \mapsto p\), where \(v \in T_pM\): the topology and the \(C^\infty\) atlas of \(TM\) is defined as this: for each chart, \((U_\alpha \subseteq M, \phi_\alpha)\), define the chart, \((\pi^{-1} (U_\alpha) \subseteq TM, \phi'_\alpha)\), where \(\phi'_\alpha: \pi^{-1} (U_\alpha) \to \phi (U_\alpha) \times \mathbb{R}^d \subseteq \mathbb{R}^{2 d} \text{ or } \mathbb{H}^{2 d}\), \(v \mapsto (\phi_\alpha (\pi (v)), \phi''_\alpha (v))\), where \(\phi''_\alpha: \pi^{-1} (U_\alpha) \to \mathbb{R}^d\), \(v \mapsto (v^1, ..., v^d)\), where \(v = v^j \partial x^j\), where \((x^1, ..., x^d)\) are the coordinates of the chart, \((U_\alpha, \phi_\alpha)\), and take \(B_\alpha := \{{\phi'_\alpha}^{-1} (U'_\alpha) \vert \forall U'_\alpha \in \{\text{ the open subsets of } \phi_\alpha (U_\alpha) \times \mathbb{R}^d\}\}\), and take \(B := \cup_{\alpha} B_\alpha\) as the topological basis
3: Note
The basis is valid according to Description 2 of some criteria for any set of open sets to be a basis, as is shown subsequently.
1) \(TM\) is obviously the union of all the elements of the basis.
2) for any \({\phi'_\alpha}^{-1} (U'_\alpha)\) and \({\phi'_\beta}^{-1} (U'_\beta)\) such that \({\phi'_\alpha}^{-1} (U'_\alpha) \cap {\phi'_\beta}^{-1} (U'_\beta) \neq \emptyset\), \(U'_\alpha = \cup_{\gamma \in A_\alpha} U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma}\) where \(A_\alpha\) is a possibly uncountable index set and \(U'_\beta = \cup_{\gamma' \in A_\beta} U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}\) where \(A_\beta\) is a possibly uncountable index set, by Note for the definition of product topology, and \({\phi'_\alpha}^{-1} (U'_\alpha) \cap {\phi'_\beta}^{-1} (U'_\beta) = (\cup_{\gamma \in A_\alpha} {\phi'_\alpha}^{-1} (U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma})) \cap (\cup_{\gamma' \in A_\beta} {\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}))\), by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets.
For any \(p \in {\phi'_\alpha}^{-1} (U'_\alpha) \cap {\phi'_\beta}^{-1} (U'_\beta)\), \(p \in {\phi'_\alpha}^{-1} (U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma}) \cap {\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'})\) for a \(\gamma \in A_\alpha\) and a \(\gamma' \in A_\beta\). We are going to show that \(S := {\phi'_\alpha}^{-1} (U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma}) \cap {\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'})\) is an element of the basis. In order for that, we are going to show that \(\phi'_\alpha (S)\) is open on \(\phi'_\alpha (\pi^{-1} (U_\alpha))\), because then, \(S\) will be the preimage of the open subset under \(\phi'_\alpha\), which exists in the basis by definition.
\(\phi'_\alpha (S) = \phi'_\alpha ({\phi'_\alpha}^{-1} (U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma})) \cap \phi'_\alpha ({\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \pi^{-1} (U_\alpha))\), by the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets, \(= (U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma}) \cap \phi'_\alpha ({\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \pi^{-1} (U_\alpha))\).
\(\phi'_\beta ({\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \pi^{-1} (U_\alpha)) = \phi'_\beta ({\phi'_\beta}^{-1} (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'})) \cap \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\), by the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets, \(= (U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\).
In fact, \(\phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta)) = \phi_\beta (U_\alpha \cap U_\beta) \times \mathbb{R}^d\), and is open on \(\mathbb{R}^{2 d}\) or \(\mathbb{H}^{2 d}\). Likewise, \(\phi'_\alpha (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\) is open on \(\mathbb{R}^{2 d}\) or \(\mathbb{H}^{2 d}\), by the symmetry.
So, \(\phi'_\alpha (S) = (U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma}) \cap \phi'_\alpha \circ {\phi'_\beta}^{-1} ((U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta)))\).
Let us think of \(\phi'_\alpha \circ {\phi'_\beta}^{-1}: \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta)) \to \phi'_\alpha (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\), obviously bijective.
Let us prove that \(\phi'_\alpha \circ {\phi'_\beta}^{-1}\) is diffeomorphic. \(\phi'_\alpha \circ {\phi'_\beta}^{-1}: (x^1, ..., x^d, v^1, ..., v^d) \mapsto (\phi_\alpha ({\phi_\beta}^{-1}) (x), \partial \phi^1_\alpha (x) / \partial x^j v^j, ..., \partial \phi^d_\alpha (x) / \partial x^j v^j)\), which is \(C^\infty\). The inverse, \((\phi'_\alpha \circ {\phi'_\beta}^{-1})^{-1} = \phi'_\beta \circ {\phi'_\alpha}^{-1}\), is likewise \(C^\infty\), by the symmetry. So, \(\phi'_\alpha \circ {\phi'_\beta}^{-1}\) is indeed diffeomorphic.
As \(\phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\) is open on \(\mathbb{R}^{2 d}\) or \(\mathbb{H}^{2 d}\), \((U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\) is open on \(\mathbb{R}^{2 d}\) or \(\mathbb{H}^{2 d}\), and so, is open on \(\phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\), and \(\phi'_\alpha \circ {\phi'_\beta}^{-1} ((U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta)))\) is open on \(\phi'_\alpha (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\), because \(\phi'_\alpha \circ {\phi'_\beta}^{-1}\) is diffeomorphic, and so, \((U''_{\alpha, \gamma} \times U'''_{\alpha, \gamma}) \cap \phi'_\alpha \circ {\phi'_\beta}^{-1} ((U''_{\beta, \gamma'} \times U'''_{\beta, \gamma'}) \cap \phi'_\beta (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta)))\) is open on \(\phi'_\alpha (\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta))\), and so, is open on \(\phi'_\alpha (\pi^{-1} (U_\alpha))\).
So, \(S\) is an element of the basis.
So, the criterion, 2), is satisfied.
The topological space is Hausdorff, because for any distinct points on \(TM\), if they are on some distinct fibers, there are some disjoint chart open subsets, \(U_\alpha, U_\beta\), and \(\pi^{-1} (U_\alpha) \cap \pi^{-1} (U_\beta) = \emptyset\); if they are on a same fiber, there are some open subsets of \(\phi'_\alpha (\pi^{-1} (U_\alpha))\), \((U_\alpha \times U') \cap (U_\alpha \times U'') = \emptyset\), because \(\mathbb{R}^d\) is Hausdorff, and \({\phi'_\alpha}^{-1} (U_\alpha \times U') \cap {\phi'_\alpha}^{-1} (U_\alpha \times U'') = \emptyset\).
The topological space is 2nd-countable, because the charts on \(M\) can be taken countably and the open subsets of \(\phi'_\alpha (\pi^{-1} (U_\alpha))\) can be taken countably, because \(\mathbb{R}^{2 d}\) or \(\mathbb{H}^{2 d}\) is 2nd-countable.
The atlas is \(C^\infty\), as has been proved above.
\(\pi\) is \(C^\infty\), because with respect to the charts, \((\pi^{-1} (U_\alpha) \subseteq TM, \phi'_\alpha)\) and \((U_\alpha \subseteq M, \phi)\), the coordinates function is \((\phi_\alpha (\pi (v)), \phi''_\alpha (v)) \mapsto \phi_\alpha (\pi (v))\), \(C^\infty\).
Any tangent vectors bundle is a kind of \(C^\infty\) vectors bundle: \(\pi\) is a locally trivial surjection of rank \(d\), as can be confirmed straightforwardly.
