2025-01-07

936: For C Manifold with Boundary, Tangent Vectors Bundle, and Immersed Submanifold with Boundary of Base Space, Tangent Vectors Bundle of Submanifold with Boundary Is Canonically C Embedded into Restricted Vectors Bundle

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description/proof of that for C manifold with boundary, tangent vectors bundle, and immersed submanifold with boundary of base space, tangent vectors bundle of submanifold with boundary is canonically C embedded into restricted vectors bundle

Topics


About: C manifold

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any C 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 C 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:
M: { the d -dimensional C manifolds with boundary }
(TM,M,π): = the tangent vectors bundle of M
S: { the d -dimensional immersed submanifolds with boundary of M}
ι: :SM, = the inclusion 
TM|S: = the restricted C vectors bundle , with π:TM|SS as the projection
(TS,S,ρ): = the tangent vectors bundle of S
ι: :TSTM|S,vdιv
//

Statements:
ι{ the C embeddings }
//


2: Note


Note that although TM|S is an immersed submanifold with boundary of TM (refer to Note for the definition of restricted C vectors bundle), it is not any embedded submanifold with boundary of TM in general, which is why the proof of this proposition is not so simple.


3: Proof


Whole Strategy: Step 1: see that ι is injective; Step 2: according to the definition of restricted C vectors bundle, for each sS, take a chart around s, (UβS,ϕβ), the induced chart, (ρ1(Uβ)TS,ϕβ~), and a chart, (π1(Uβ)TM|S,ϕβ), and see what the components function of ι is like; Step 3: see that ι is C; Step 4: see what dι is like the components-wise, and see that dι is injective on each fiber; Step 5: see that the codomain restriction of ι, ι:TSι(TS)TM|S is homeomorphic; Step 6: conclude the proposition.

Step 1:

Let us see that ι is injective.

As ι is a C immersion, dι:TSTM is injective.

Although the codomain of ι is TM|S instead of TM, the injectiveness is only about the map between-sets-wise, and the injectiveness of ι holds.

Step 2:

Let us take a chart for TS and a chart for TM|S in order to study the local behaviors of ι via the coordinates function.

Let sS be any.

According to the definition of restricted C vectors bundle, we can take a chart, (UβS,ϕβ), and a chart, (π1(Uβ)TM|S,ϕβ).

ϕβ is induced by a trivialization for TM, Φβ:π1(Uβ)Uβ×Rd, where UβM is a trivializing open subset. In fact, we can take Uβ as the domain of a chart, (UβM,ϕβ), and we do so, and we can take the trivialization as the canonical one induced by the chart, and we do so.

Let us take the canonical chart induced by (UβS,ϕβ), (ρ1(Uβ)TS,ϕβ~).

Let us take the canonical chart induced by (UβM,ϕβ), (π1(Uβ)TM,ϕβ~).

ι(ρ1(Uβ))π1(Uβ), because ι is fiber-preserving, because dι is so.

So, we are going to study the components of ι with respect to (ρ1(Uβ)TS,ϕβ~) and (π1(Uβ)TM|S,ϕβ).

In fact, the components function is of the form, :(v1,...,vd,x1,...,xd)(w1,...,wd,x1,...,xd), because the both charts use the same ϕβ.

Step 3:

As a comparison, let us think of the global differential of ι, dι:TSTM.

The components function of dι with respect to (ρ1(Uβ)TS,ϕβ~) and (π1(Uβ)TM,ϕβ~), is of the form, :(v1,...,vd,x1,...,xd)(w1,...,wd,y1,...,yd), whose point is that (w1,...,wd) is exactly the same with the components function of ι, which is because while both ι and dι do the mappings with the same dι, both (π1(Uβ)TM|S,ϕβ) and (π1(Uβ)TM,ϕβ~) are induced by the same trivialization, Φβ.

dι is C, by the proposition that for any C map between any C manifolds with boundary, the global differential is C.

That means that wj is a C function of (v1,...,vd,x1,...,xd).

Then, also :(v1,...,vd,x1,...,xd)(w1,...,wd,x1,...,xd) is C.

So, ι is C.

Step 4:

There are the canonical chart for TTS induced by (ρ1(Uβ)TS,ϕβ~) and the canonical chart for TTM|S induced by (π1(Uβ)TM|S,ϕβ).

As is well-known, the components function of dι with respect to the charts is of the form, :(t1,...,t2d,x1,...,xd,v1,...,vd)(jx1tj,...,jxdtj,jw1tj,...,jwdtj,x1,...,xd,w1,...,wd), where j is by xj for j{1,...,d} and by vjd for j{d+1,...,2d}, =(t1,...,td,jw1tj,...,jwdtj,x1,...,xd,w1,...,wd).

As a comparison, let us think of ddι.

There is the canonical chart for TTM induced by (π1(Uβ)TM,ϕβ~).

The components function of ddι with respect to the chart for TTS and the chart for TTM is of the form, :(t1,...,t2d,x1,...,xd,v1,...,vd)(jy1tj,...,jydtj,jw1tj,...,jwdtj,y1,...,yd,w1,...,wd), where j is by xj for j{1,...,d} and by vjd for j{d+1,...,2d}, but as y depends only on x, jyltj s are only for j{1,...,d}.

ddι is injective on each fiber, because dι is a C immersion, by the proposition that for any C immersion between any C manifolds with boundary, its global differential is a C immersion.

That means that the matrix, (y1/x1...y1/xd0...0...yd/x1...yd/xd0...0w1/x1...w1/xdw1/v1...w1/vd...wd/x1...wd/xdwd/v1...wd/vd), is rank 2d.

That means that there is a 2d×2d submatrix, (yj1/x1...yj1/xd0...0...yjk/x1...yjk/xd0...0wl1/x1...wl1/xdwl1/v1...wl1/vd...wlm/x1...wlm/xdwlm/v1...wlm/vd), that is invertible.

In fact, kd, because otherwise, the submatrix would not be invertible by 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: if d<k, 2dk<d, but as the k rows had the d same columns 0, the submatrix would not be invertible.

Each of the top k rows can be changed to have a 1 1 component and 0 others without any duplication to keep the matrix invertible, by 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.

That is a 2d×2d submatrix of the matrix for the components function of dι, which means that the matrix is rank 2d, which implies that dι is injective on each fiber.

So, ι is a C immersion.

Step 5:

{ρ1(Uβ)|βB} is an open cover of TS.

Let Φβ:π1(Uβ)Uβ×Rd be the trivialization for TM|S defined in the definition of restricted C vectors bundle.

Let Φβ:ρ1(Uβ)Uβ×Rd be the canonical trivialization for TS.

Let us think of Φβι|ρ1(Uβ)Φβ1:Uβ×RdUβ×Rd.

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, ι|ρ1(Uβ)=Φβ1Φβι|ρ1(Uβ)Φβ1Φβ:ρ1(Uβ)π1(Uβ) is a continuous embedding: Φβ and Φβ are some homeomorphisms. Also the codomain expansion, ι|ρ1(Uβ):ρ1(Uβ)TM|S, is a continuous embedding, by the proposition that any expansion of any continuous embedding on the codomain is a continuous embedding.

ι|ρ1(Uβ)(ρ1(Uβ))=π1(Uβ)ι(TS), which is open on TM|Sι(TS), because π1(Uβ) is open on TM|S.

So, ι is a continuous embedding, by 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.

That means that the codomain restriction of ι, ι:TSι(TS)TM|S, is a homeomorphism.

Step 6:

So, ι is an injective immersion whose codomain restriction is a homeomorphism, which means that ι is a C embedding.


References


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