definition of \(C^\infty\) \((p, q)\)-tensors bundle over \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \((p, q)\)-tensors space at point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
Target Context
- The reader will have a definition of \(C^\infty\) \((p, q)\)-tensors 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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\( p\): \(\in \mathbb{N}\)
\( q\): \(\in \mathbb{N}\)
\( T^p_q (TM)\): \(= \biguplus_{m \in M} T^p_q (T_mM)\), denoted also as \(TM \otimes ... \otimes TM \otimes TM^* \otimes ... \otimes TM^*\)
\( \pi\): \(: T^p_q (TM) \to M, t \mapsto m\), where \(t \in T^p_q (T_mM)\)
\(*(T^p_q (TM), M, \pi)\):
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Conditions:
\((T^p_q (TM), M, \pi)\) is the \(C^\infty\) vectors bundle constructed with the following to-be-set-of-local-trivializations: for any covering set of charts, \(\{(U_j \subseteq M, \phi_j)\}\), \(\Phi_j: \pi^{-1} (U_j) \to U_j \times \mathbb{R}^{d^{(p + q)}}, t = t^{j_1, ..., j_p}_{l_1, ..., l_q} [((\partial / \partial x^{j_1}, ..., \partial / \partial x^{j_p}, d x^{l_1}, ..., d x^{l_q}))] \mapsto (\pi (t), (t^{1, ..., 1}_{1, ..., 1}, ..., t^{d, ..., d}_{d, ..., d}))\), by the proposition that for any \(C^\infty\) manifold with boundary and any real vectors space at each point with any same dimension, \(k\), any to-be-set-of-local-trivializations determines the canonical \(C^\infty\) vectors bundle of rank \(k\)
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\(T^p_q (TM)\) is very often implicitly identified with \(\biguplus_{m \in M} L (T_mM^*, ..., T_mM^*, T_mM, ..., T_mM: \mathbb{R})\) by the canonical bijection, \(f': T^p_q (TM) \to \biguplus_{m \in M} L (T_mM^*, ..., T_mM^*, T_mM, ..., T_mM: \mathbb{R}), t \in T^p_q (T_mM) \mapsto f (t)\) where \(f\) is the canonical 'vectors spaces - linear morphisms' isomorphism mentioned in the definition of canonical 'vectors spaces - linear morphisms' isomorphism between tensors space at point on \(C^\infty\) manifold with boundary and tensors space with respect to real numbers field and \(p\) cotangent vectors spaces and \(q\) tangent vectors spaces and field.
So, although each \(t = t^{j_1, ..., j_p}_{l_1, ..., l_q} [((\partial / \partial x^{j_1}, ..., \partial / \partial x^{j_p}, d x^{l_1}, ..., d x^{l_q}))] \in T^p_q (TM)\) is not really any multilinear map, \(f (t) = t^{j_1, ..., j_p}_{l_1, ..., l_q} \widetilde{\partial / \partial x^{j_1}} \otimes ... \otimes \widetilde{\partial / \partial x^{j_p}} \otimes d x^{l_1} \otimes ... \otimes d x ^{l_q}\) is very often implicitly used instead of \(t\).
2: Note
Let us see that the definition indeed conforms to the requirements for the proposition that for any \(C^\infty\) manifold with boundary and any real vectors space at each point with any same dimension, \(k\), any to-be-set-of-local-trivializations determines the canonical \(C^\infty\) vectors bundle of rank \(k\).
\(T^p_q (T_mM)\) is a \(d^{(p + q)}\)-dimensional \(\mathbb{R}\) vectors space, by the proposition that the tensor product of any \(k\) finite-dimensional vectors spaces has the basis that consists of the classes induced by any basis elements: \(T_mM\) is \(d\)-dimensional and \(T_mM^*\) is \(d\)-dimensional, as is well-known.
For each \(m \in U_j\), \(\Phi_j \vert_{T^p_q (T_mM)}: T^p_q (T_mM) \to \{m\} \times \mathbb{R}^{d^{(p + q)}}, t = t^{j_1, ..., j_p}_{l_1, ..., l_q} [((\partial / \partial x^{j_1}, ..., \partial / \partial x^{j_p}, d x^{l_1}, ..., d x^{l_q}))] \mapsto (\pi (t), (t^{1, ..., 1}_{1, ..., 1}, ..., t^{d, ..., d}_{d, ..., d}))\) is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that between any vectors spaces, any map that maps any basis onto any basis bijectively and expands the mapping linearly is a 'vectors spaces - linear morphisms' isomorphism.
\(\begin{pmatrix} T_{j, l} \end{pmatrix}\) is \(\begin{pmatrix} \partial (\phi_l \circ {\phi_j}^{-1})^{j_1} / \partial x^{m_1} ... \partial (\phi_l \circ {\phi_j}^{-1})^{j_p} / \partial x^{m_p} \partial (\phi_j \circ {\phi_l}^{-1})^{n_1} / \partial x'^{l_1} ... \partial (\phi_j \circ {\phi_l}^{-1})^{n_q} / \partial x'^{l_q} \end{pmatrix}\) where \(\{(j_1, ..., j_p, l_1, ..., l_q)\}\) constitutes the index for \(l\) and \(\{(m_1, ..., m_p, n_1, ..., n_q)\}\) constitutes the index for \(j\), by the proposition that for any \(C^\infty\) manifold with boundary and the \((p, q)\)-tensors space at any point, the transition of the components of any tensor with respect to the standard bases with respect to any charts is this.
\(\begin{pmatrix} T_{j, l} \end{pmatrix}\) is \(C^\infty\), because \(\phi_l \circ {\phi_j}^{-1}\) and \(\phi_j \circ {\phi_l}^{-1}\) are \(C^\infty\): while \(\partial (\phi_l \circ {\phi_j}^{-1})^{j_1} / \partial x^{m_1} ... \partial (\phi_l \circ {\phi_j}^{-1})^{j_p} / \partial x^{m_p}\) is \(C^\infty\) as a map from \(\phi_j (U_j \cap U_l)\) and \(\partial (\phi_j \circ {\phi_l}^{-1})^{n_1} / \partial x'^{l_1} ... \partial (\phi_j \circ {\phi_l}^{-1})^{n_q} / \partial x'^{l_q}\) is \(C^\infty\) as a map from \(\phi_l (U_j \cap U_l)\), \(\partial (\phi_l \circ {\phi_j}^{-1})^{j_1} / \partial x^{m_1} ... \partial (\phi_l \circ {\phi_j}^{-1})^{j_p} / \partial x^{m_p} \partial (\phi_j \circ {\phi_l}^{-1})^{n_1} / \partial x'^{l_1} ... \partial (\phi_j \circ {\phi_l}^{-1})^{n_q} / \partial x'^{l_q}\) is \(C^\infty\) as a map from \(U_j \cap U_l\), because \(\phi_j \vert_{U_j \cap U_l}: U_j \cap U_l \to \phi_j (U_j \cap U_l)\) and \(\phi_l \vert_{U_j \cap U_l}: U_j \cap U_l \to \phi_l (U_j \cap U_l)\) are \(C^\infty\).
So, the definition indeed conforms to the requirements for the proposition that for any \(C^\infty\) manifold with boundary and any real vectors space at each point with any same dimension, \(k\), any to-be-set-of-local-trivializations determines the canonical \(C^\infty\) vectors bundle of rank \(k\).
Let us see that the topology and the \(C^\infty\) atlas of \(T^p_q (TM)\) do not depend on the choice of \(\{(U_j \subseteq M, \phi_j)\}\), by the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.
A set of covering charts by \(\{(U_j \subseteq M, \phi_j)\}\) is \(\{(\pi^{-1} (U_m) \subseteq T^p_q (TM), \widetilde{\phi_m})\}\), where \(m \in U_m \subseteq U_j\).
Let another choice be \(\{(U'_{j'} \subseteq M, \phi'_{j'})\}\).
A set of covering charts by \(\{(U'_{j'} \subseteq M, \phi'_{j'})\}\) is \(\{(\pi^{-1} (U'_{m'}) \subseteq T^p_q (TM), \widetilde{\phi'_{m'}})\}\), where \(m' \in U'_{m'} \subseteq U'_{j'}\).
\(\{\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'})\}\) is a common chart domains open cover, because \((\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'}) \subseteq T^p_q (TM), \widetilde{\phi_m} \vert_{\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'})})\) and \((\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'}) \subseteq T^p_q (TM), \widetilde{\phi'_{m'}} \vert_{\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'})})\) are some charts.
The transition is \(\widetilde{\phi'_{m'}} \vert_{\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'})} \circ (\widetilde{\phi_m} \vert_{\pi^{-1} (U_m) \cap \pi^{-1} (U'_{m'})})^{-1} = \lambda \circ (\phi'_{m'}, id) \circ \Phi'_{j'} \circ {\Phi_j}^{-1} \circ ({\phi_m}^{-1}, id^{-1}) \circ \lambda^{-1}\): strictly speaking, each constituent needs to be appropriately restricted, but let us omit the marks of the restrictions here.
\(\Phi'_{j'} \circ {\Phi_j}^{-1}\) is a diffeomorphism, because it is essentially no different from \(\Phi_l \circ {\Phi_j}^{-1}\).
The transition is a diffeomorphism, because \(\lambda\), \((\phi'_{m'}, id)\), \(\Phi'_{j'} \circ {\Phi_j}^{-1}\), \(({\phi_m}^{-1}, id^{-1})\), and \(\lambda^{-1}\) are some diffeomorphisms.
So, the topology and the \(C^\infty\) atlas of \(T^p_q (TM)\) do not depend on the choice of \(\{(U_j \subseteq M, \phi_j)\}\).
