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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( m\): \(\in M\)
\( T_mM\): \(= \text{ the tangent vectors space at } m\)
\( T_mM^*\): \(= \text{ the covectors space of } T_mM\)
\( p\): \(\in \mathbb{N}\)
\( q\): \(\in \mathbb{N}\)
\(*T^p_q (T_mM)\):
//
Conditions:
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\(T^p_q (T_mM)\) is regarded to be as in the definition of tensors space with respect to field and \(k\) vectors spaces and vectors space over field or as in the definition of tensor product of \(k\) vectors spaces over field.
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 tensors space with respect to field and \(k\) vectors spaces and vectors space over field or in Note for the definition of tensor product of \(k\) vectors spaces over field.
\(T^1_0 (T_mM)\) is the tangent vectors space.
\(T^0_1 (T_mM)\) is called "cotangent vectors space".
