definition of pullback of \((0, q)\)-tensors at point by \(C^\infty\) map between \(C^\infty\) manifolds with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of pullback of \((0, q)\)-tensors at point by \(C^\infty\) map between \(C^\infty\) manifolds 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:
\( M_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( M_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\( m\): \(\in M_1\)
\( q\): \(\in \mathbb{N}\)
\(*f^*_m\): \(: T^0_q (T_{f (m)}M_2) \to T^0_q (T_mM_1)\), \(\in \{\text{ the linear maps }\}\)
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Conditions:
when \(q = 0\), \(\forall t \in T^0_q (T_{f (m)}M_2) (f^*_m (t) = t \circ f (m))\)
when \(0 \lt q\), \(\forall t \in T^0_q (T_{f (m)}M_2), \forall v_1, ..., v_q \in T_mM_1 (f^*_m (t) (v_1, ..., v_q) = t (d f_m (v_1), ..., d f_m (v_q)))\)
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2: Note
Let us see that \(f^*_m\) is indeed into \(T^0_q (T_mM_1)\).
When \(q = 0\), \(f^*_m (t) = t \circ f (m) \in \mathbb{R}\).
Let us suppose that \(0 \lt q\).
\(f^*_m (t) (v_1, ..., r v_j + r' v'_j, ..., v_q) = t (d f_m (v_1), ..., d f_m (r v_j + r' v'_j), ..., d f_m (v_q)) = t (d f_m (v_1), ..., r d f_m (v_j) + r' d f_m (v'_j), ..., d f_m (v_q))\), because \(d f_m\) is linear, \(= r t (d f_m (v_1), ..., d f_m (v_j), ..., d f_m (v_q)) + r' t (d f_m (v_1), ..., d f_m (v'_j), ..., d f_m (v_q))\), because \(t\) is multilinear, \(= r f^*_m (t) (v_1, ..., v_j, ..., v_q) + r' f^*_m (t) (v_1, ..., v'_j, ..., v_q)\).
Let us see that \(f^*_m\) is indeed linear.
When \(q = 0\), \(f^*_m (r t + r' t') = r f^*_m (t) + r' f^*_m (t')\), because \(f^*_m (r t + r' t') = (r t + r' t') \circ f (m) = r t \circ f (m) + r' t' \circ f (m) = r f^*_m (t) + r' f^*_m (t')\).
Let us suppose that \(0 \lt q\).
\(f^*_m (r t + r' t') = r f^*_m (t) + r' f^*_m (t')\), because \(f^*_m (r t + r' t') (v_1, ..., v_q) = (r t + r' t') (d f_m (v_1), ..., d f_m (v_q)) = r t (d f_m (v_1), ..., d f_m (v_q)) + r' t' (d f_m (v_1), ..., d f_m (v_q)) = r f^*_m (t) (v_1, ..., v_q) + r' f^*_m (t') (v_1, ..., v_q) = (r f^*_m (t) + r' f^*_m (t')) (v_1, ..., v_q)\).
