definition of \((p, q)\)-tensors 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 tensors space with respect to field and \(k\) vectors spaces and vectors space over field.
- The reader knows a definition of tensor product of \(k\) vectors spaces over field.
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of \((p, q)\)-tensors 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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( m\): \(\in M\)
\( T_mM\): \(= \text{ the tangent vectors space at } m\)
\( T_mM^*\): \(= \text{ the covectors space of } T_mM\)
\( p\): \(\in \mathbb{N}\)
\( q\): \(\in \mathbb{N}\)
\(*T^p_q (T_mM)\): \(= T_mM \otimes ... \otimes T_mM \otimes T_mM^* \otimes ... \otimes T_mM^*\) where \(T_mM\) appears \(p\) times and \(T_mM^*\) appears \(q\) times when \(p \neq 0\) or \(q \neq 0\); \(= \mathbb{R}\) when \(p = q = 0\), \(\in \{\text{ the } \mathbb{R} \text{ vectors spaces }\}\)
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Conditions:
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2: Note
\(T_mM\) 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, \(T_mM^*\) is an \(\mathbb{R}\) vectors space as is shown in Note for the definition of tensors space with respect to field and \(k\) vectors spaces and vectors space over field, and \(T^p_q (T_mM)\) is indeed an \(\mathbb{R}\) vectors space as is shown in Note for the definition of tensor product of \(k\) vectors spaces over field.
\(T^p_q (T_mM)\) is canonically 'vectors spaces - linear morphisms' isomorphic to \(L (T_mM^*, ..., T_mM^*, T_mM, ..., T_mM: \mathbb{R})\), by Note for 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, and quite often, the 2 spaces are implicitly identified by the isomorphism, which is the reason why \(T^p_q (T_mM)\) is called "tensors space".
\(T^1_0 (T_mM)\) is the tangent vectors space.
\(T^0_1 (T_mM)\) is called "cotangent vectors space".
