description/proof of that for \(C^\infty\) map from \(C^\infty\) manifold with boundary into finite-product \(C^\infty\) manifold with boundary, composition of canonical 'vectors spaces - linear morphisms' isomorphism after differential is direct sum of differentials of compositions of projections after map
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of finite-product \(C^\infty\) manifold with boundary.
- The reader knows a definition of differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
- The reader admits the proposition that for any finite-product \(C^\infty\) manifold with boundary and any point, there is the 'vectors spaces - linear morphisms' isomorphism from the tangent vectors space at the point onto the direct sum of the tangent vectors spaces of the constituent manifolds with boundary at the corresponding points as the para-product of the differentials of the projections.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) map from any \(C^\infty\) manifold with boundary into any finite-product \(C^\infty\) manifold with boundary, the composition of the canonical 'vectors spaces - linear morphisms' isomorphism after the differential is the direct sum of the differentials of the compositions of the projections after the map.
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:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(\{M_1, ..., M_{n - 1}\}\): \(\subseteq \{\text{ the } C^\infty \text{ manifolds }\}\)
\(M_n\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_1 \times ... \times M_n\): \(= \text{ the product } C^\infty \text{ manifold with boundary }\)
\(f\): \(: M \to M_1 \times ... \times M_n\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\(\{\pi_1, ..., \pi_n\}\): \(\pi_j: M_1 \times ... \times M_n \to M_j = \text{ the projection }\)
\(d f_m\): \(: T_mM \to T_{f (m)}(M_1 \times ... \times M_n)\), \(= \text{ the differential at } m\)
\(g\): \(: T_{f (m)}(M_1 \times ... \times M_n) \to T_{f (m)^1}M_1 \oplus ... \oplus T_{f (m)^n}M_n, v \mapsto (d {\pi_1}_{f (m)} v, ..., d {\pi_n}_{f (m)} v)\), \(= \text{ the canonical 'vectors spaces - linear morphisms' isomorphism }\)
\(g \circ d f_m\): \(: T_mM \to T_{f (m)^1}M_1 \oplus ... \oplus T_{f (m)^n}M_n\)
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Statements:
\(g \circ d f_m = (d (\pi_1 \circ f)_m, ..., d (\pi_n \circ f)_m)\)
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2: Proof
Whole Strategy: Step 1: see that \(g \circ d f_m = (d {\pi_1}_{f (m)} \circ d f_m, ..., d {\pi_n}_{f (m)} \circ d f_m)\).
Step 1:
As we already know that \(g: v \mapsto (d {\pi_1}_{f (m)} v, ..., d {\pi_n}_{f (m)} v)\), by the proposition that for any finite-product \(C^\infty\) manifold with boundary and any point, there is the 'vectors spaces - linear morphisms' isomorphism from the tangent vectors space at the point onto the direct sum of the tangent vectors spaces of the constituent manifolds with boundary at the corresponding points as the para-product of the differentials of the projections, for each \(v \in T_mM\), \(g \circ d f_m (v) = (d {\pi_1}_{f (m)} \circ d f_m (v), ..., d {\pi_n}_{f (m)} \circ d f_m (v)) = (d (\pi_1 \circ f)_m (v), ..., d (\pi_n \circ f)_m (v))\), as is well-known.
That means that \(g \circ d f_m = (d (\pi_1 \circ f)_m, ..., d (\pi_n \circ f)_m)\).
