definition of \(C^\infty\) \(q\)-covectors 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 \(q\)-covectors 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 \(C^\infty\) \(q\)-covectors 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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\( q\): \(\in \mathbb{N}\)
\( \Lambda_q (TM)\): \(= \biguplus_{m \in M} \Lambda_q (T_mM)\)
\( \pi\): \(: \Lambda_q (TM) \to M, v \mapsto m\), where \(v \in \Lambda_q (T_mM)\)
\(*(\Lambda_q (TM), M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundle of rank } _dC_q\}\)
//
Conditions:
\((\Lambda_q (TM), M, \pi) \in \{\text{ the } C^\infty \text{ subbundles of } (T^0_q (TM), M, \pi)\}\)
//
\(\Lambda_q (TM)\) is very often implicitly identified with \(\biguplus_{m \in M} \Lambda_q (T_mM: \mathbb{R})\) by the canonical bijection, \(f': \Lambda_q (TM) \to \biguplus_{m \in M} \Lambda_q (T_mM: \mathbb{R}), t \in \Lambda_q (T_mM) \mapsto f (t)\) 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 \(t \in \Lambda_q (TM)\) is not really any multilinear map, \(f' (t)\) is very often implicitly used instead of \(t\).
2: Note
\((\Lambda_q (TM), M, \pi)\)'s being a subbundle of \((T^0_q (TM), M, \pi)\) means that \(\Lambda_q (TM)\) has the subspace topology of \(T^0_q (TM)\) and the adopting atlas of \(T^0_q (TM)\) (which means that \(\Lambda_q (TM)\) is the embedding submanifold with boundary of \(T^0_q (TM)\)).
Let us see that \((\Lambda_q (TM), M, \pi)\) is indeed a subbundle of \((T^0_q (TM), M, \pi)\), by the proposition that for any \(C^\infty\) vectors bundle, the union of any \(k\)-dimensional vectors subspaces of the fibers that allows any local \(C^\infty\) frames is a \(C^\infty\) vectors subbundle with the subspace topology and the adopting atlas.
Note that in the following argument, we implicitly identify \(\Lambda_q (TM)\) with \(\biguplus_{m \in M} \Lambda_q (T_mM: \mathbb{R})\): for example, \(d x^{j_1} \wedge ... \wedge d x^{j_q}\) is really on \(\biguplus_{m \in M} \Lambda_q (T_mM: \mathbb{R})\), and we really mean \(f'^{-1} (d x^{j_1} \wedge ... \wedge d x^{j_q})\), but such exactitude is too hard to be maintained (and is almost never seen in the literature).
\(\Lambda_q (T_mM)\) is a \(_dC_q\)-dimensional vectors subspace of \(T^0_q (T_mM)\).
For each \(m \in M\), let us take any chart, \((U_m \subseteq M, \phi_m)\).
\(d x^j: U_m \to T^0_1 (TM)\) is a local \(C^\infty\) section for \((T^0_1 (TM), M, \pi)\).
For any \(j_1 \lt ... \lt j_q\), \(d x^{j_1} \wedge ... \wedge d x^{j_q}\) is a local \(C^\infty\) section for \((T^0_q (TM), M, \pi)\), because \(= \sum_{\sigma \in S_q} sgn \sigma d x^{j_{\sigma_1}} \otimes ... \otimes d x^{j_{\sigma_q}}\) , by the proposition that the wedge product of any 1-covectors is the sum of the signed reordered tensor products of the 1-covectors, and each \(d x^{j_{\sigma_1}} \otimes ... \otimes d x^{j_{\sigma_q}}\) is \(C^\infty\).
\(\{d x^{j_1} \wedge ... \wedge d x^{j_q} \vert j_1 \lt ... \lt j_q\}\) is a required set of \(_dC_q\) local \(C^\infty\) sections: for each \(m' \in U_m\), \(\{d x^{j_1} \vert_{m'} \wedge ... \wedge d x^{j_q} \vert_{m'} \vert j_1 \lt ... \lt j_q\}\) is a basis for \(\Lambda_q (T_mM: \mathbb{R})\), by the proposition that the \(q\)-covectors space of any vectors space has the basis that consists of the wedge products of the increasing elements of the dual basis of the vectors space.
