definition of \(q\)-covectors space at point on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vectors space at point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of antisymmetric tensors space with respect to field and \(k\) same vectors spaces and vectors space over field.
- The reader knows a definition of \((p, q)\)-Tensors Space at Point on \(C^\infty\) Manifold with Boundary.
- The reader knows a definition of canonical 'vectors spaces - linear morphisms' isomorphism between \((p, q)\)-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.
Target Context
- The reader will have a definition of \(q\)-covectors space at point on \(C^\infty\) 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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( m\): \(\in M\)
\( T_mM\): \(= \text{ the tangent vectors space at } m\)
\( q\): \(\in \mathbb{N}\)
\( \Lambda_q (T_mM: \mathbb{R})\): \(= \text{ the } q \text{ -covectors space }\)
\( f\): \(: T^0_q (T_mM) \to L (T_mM, ..., T_mM: \mathbb{R})\), \(= \text{ the canonical 'vectors spaces - linear morphisms' isomorphism }\)
\(*\Lambda_q (T_mM)\): \(= f^{-1} (\Lambda_q (T_mM: \mathbb{R}))\) when \(q \neq 0\); \(= \mathbb{R}\) when \(q = 0\), \(\in \{\text{ the } \mathbb{R} \text{ vectors spaces }\}\)
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Conditions:
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2: Note
Another definition defines "\(\Lambda_q (T_mM) := \Lambda_q (T_mM: \mathbb{R})\)", but that would make \(\Lambda_q (T_mM)\) non-subspace of \(T^0_q (T_mM)\), because we have defined \(T^0_q (T_mM) := T_mM^* \otimes ... \otimes T_mM^*\), not \(= L (T_mM, ..., T_mM: \mathbb{R})\).
\(T_mM\) is an \(\mathbb{R}\) vectors space as is shown in Note for the definition of tangent vectors space at point on \(C^\infty\) manifold with boundary and \(\Lambda_q (T_mM: \mathbb{R})\) is indeed a vectors subspace of \(L (T_mM, ..., T_mM: \mathbb{R})\) as is shown in Note for the definition of antisymmetric tensors space with respect to field and \(k\) same vectors spaces and vectors space over field. So, \(f^{-1} (\Lambda_q (T_mM: \mathbb{R}))\) is a vectors subspace of \(T^0_q (T_mM)\).
\(\Lambda_q (T_mM)\) is canonically 'vectors spaces - linear morphisms' isomorphic to \(\Lambda_q (T_mM: \mathbb{R})\), and quite often, the 2 spaces are implicitly identified by the isomorphism, which is the reason why \(\Lambda_q (T_mM)\) is called "\(q\)-covectors space".
\(\Lambda_0 (T_mM) = T^0_0 (T_mM)\).
\(\Lambda_1 (T_mM) = T^0_1 (T_mM)\), because each element of \(T^0_1 (T_mM)\) is antisymmetric.
