description/proof of that for
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The table of contents of this article
Starting Context
-
The reader knows a definition of tangent vectors bundle over
manifold with boundary. -
The reader knows a definition of immersed submanifold with boundary of
manifold with boundary. -
The reader knows a definition of restricted
vectors bundle. -
The reader knows a definition of
embedding. -
The reader admits the proposition that for any
map between any manifolds with boundary, the global differential is . -
The reader admits the proposition that for any
immersion between any manifolds with boundary, its global differential is a immersion. - The reader admits the proposition that for any n x n matrix, if there are any m rows with more than any n - m same columns 0, the matrix is not invertible.
- The reader admits the proposition that for any invertible square matrix, from the top row downward through any row, each row can be changed to have a 1 1 component and 0 others without any duplication to keep the matrix invertible.
- The reader admits the proposition that in order to check the continuousness of any map between any topological spaces, the preimages of only any basis or any subbasis are enough to be checked.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that for any topological space and its 2 products with any Euclidean topological spaces, any injective continuous map between the products fiber-preserving and linear on each fiber is a continuous embedding.
- The reader admits the proposition that any expansion of any continuous embedding on the codomain is a continuous embedding.
- The reader admits the proposition that any injective map between any topological spaces is a continuous embedding if the domain restriction of the map on each element of any open cover is any continuous embedding onto any open subset of the range or the codomain.
Target Context
-
The reader will have a description and a proof of the proposition that for any
manifold with boundary, its tangent vectors bundle, and any immersed submanifold with boundary of the base space, the tangent vectors bundle of the submanifold with boundary is canonically embedded into the restricted vectors bundle.
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:
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Statements:
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2: Note
Note that although
3: Proof
Whole Strategy: Step 1: see that
Step 1:
Let us see that
As
Although the codomain of
Step 2:
Let us take a chart for
Let
According to the definition of restricted
Let us take the canonical chart induced by
Let us take the canonical chart induced by
So, we are going to study the components of
In fact, the components function is of the form,
Step 3:
As a comparison, let us think of the global differential of
The components function of
That means that
Then, also
So,
Step 4:
There are the canonical chart for
As is well-known, the components function of
As a comparison, let us think of
There is the canonical chart for
The components function of
That means that the matrix,
That means that there is a
In fact,
Each of the top
That is a
So,
Step 5:
Let
Let
Let us think of
It is an injective continuous map fiber-preserving and linear on each fiber, and so, is a continuous embedding, by the proposition that for any topological space and its 2 products with any Euclidean topological spaces, any injective continuous map between the products fiber-preserving and linear on each fiber is a continuous embedding.
So,
So,
That means that the codomain restriction of
Step 6:
So,