2025-06-16

1157: J-Half-Slice of Chart Domain w.r.t. Point

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definition of J-half-slice of chart domain w.r.t. point

Topics


About: C manifold

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of J-half-slice of chart domain with respect to point.

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 }
(UM,ϕ): { the charts for M}
J: {1,...,d}, =(j1,...,jd)
u: U such that ϕ(u)jd=0
HJ,ϕ(u)(Rd): ={rRd|j{1,...,d}J(rj=ϕ(u)j)0rjd}
HJ,u(U): U
//

Conditions:
ϕ(HJ,u(U))=ϕ(U)HJ,ϕ(u)(Rd)
//


2: Note


HJ,u(U) inevitably exists as a nonempty subset for any J and any u (as far as ϕ(u)jd=0 is satisfied, which requires that the chart is chosen to allow such a u), because ϕ(u)ϕ(U)HJ,ϕ(u)(Rd) and HJ,u(U)=ϕ1(ϕ(U)HJ,ϕ(u)(Rd)).

HJ,ϕ(u)(Rd) is like {(ϕ(u)1,...,xj1,...,xjd,...,ϕ(u)d)|jlJ{jd}(xjlR),xjdH}, where whether the 1st component is really ϕ(u)1 or xj1 and whether the last component is really ϕ(u)d or xjd depend on J.

Let πJ:RdRd,(x1,...,xj1,...,xjd,...,xd)(xj1,...,xjd) be the projection, where whether the 1st component is really x1 or xj1 and whether the last component is really xd or xjd depend on J.

πJ|HJ,ϕ(u)(Rd):HJ,ϕ(u)(Rd)RdHd is a diffeomorphism, because it is bijective and C (there are the standard chart for Rd and the standard chart for Hd such that the components function is C, because the components function has the C extension, πJ:RdRd), and the inverse is C (there are the standard chart for Hd and the standard chart for Rd such that the components function is C, because the components function has the C extension, the inverse of πJ|SJ,ϕ(u)(Rd):RdRd). Especially, πJ|HJ,ϕ(u)(Rd) is a homeomorphism.

When dJ, πJ|HJ,ϕ(u)(Rd)Hd:HJ,ϕ(u)(Rd)HdHdHd is a diffeomorphism: it is like :(ϕ(u)1,...,xj1,...,xjd)(xj1,...,xjd) where 0xjd, which is bijective and C (there are the standard chart for Hd and the standard chart for Hd such that the components function is C, because the components function has the C extension, πJ:RdRd), and the inverse is C (there are the standard chart for Hd and the standard chart for Hd such that the components function is C, because the components function has the C extension, the inverse of πJ|SJ,ϕ(u)(Rd):RdRd). Especially, πJ|HJ,ϕ(u)(Rd)Hd is a homeomorphism.

When dJ and 0ϕ(u)d (we do not need the ϕ(u)d<0 case, because we need this only for when the chart is a boundary chart, which guaranteed that 0ϕ(u)d), πJ|HJ,ϕ(u)(Rd)Hd:HJ,ϕ(u)(Rd)HdHdHd is a diffeomorphism: it is like :(ϕ(u)1,...,xj1,...,xjd,...,ϕ(u)d)(xj1,...,xjd) where 0xjd, which is bijective and C (there are the standard chart for Hd and the standard chart for Hd such that the components function is C, because the components function has the C extension, πJ:RdRd), and the inverse is C (there are the standard chart for Hd and the standard chart for Hd such that the components function is C, because the components function has the C extension, the inverse of πJ|SJ,ϕ(u)(Rd):RdRd). Especially, πJ|HJ,ϕ(u)(Rd)Hd is a homeomorphism. The condition, 0ϕ(u)d, is necessary, because otherwise, HJ,ϕ(u)(Rd)Hd=, which would not be homeomorphic to Hd.

Let us suppose that the chart is an interior chart.

ϕ(U) is an open subset of Rd.

So, ϕ(HJ,u(U)) is an open subset of HJ,ϕ(u)(Rd)Rd.

So, πJ(ϕ(HJ,u(U)))=πJ|HJ,ϕ(u)(Rd)(ϕ(HJ,u(U))) is an open subset of Hd.

πJϕ|HJ,u(U):HJ,u(U)UπJ(ϕ(HJ,u(U)))Hd is a homeomorphism, because it is πJ|ϕ(U)HJ,ϕ(u)(Rd)ϕ|HJ,u(U), and ϕ|HJ,u(U):HJ,u(U)Uϕ(U)HJ,ϕ(u)(Rd)ϕ(U) and πJ|ϕ(U)HJ,ϕ(u)(Rd):ϕ(U)HJ,ϕ(u)(Rd)HJ,ϕ(u)(Rd)πJ(ϕ(HJ,u(U)))Hd are some homeomorphisms as some restrictions of homeomorphic ϕ and πJ|HJ,ϕ(u)(Rd): ϕ(U)HJ,ϕ(u)(Rd) as the codomain of ϕ|HJ,u(U) is the subspace of ϕ(U), but as ϕ(U) is the subspace of Rd, the codomain is the subspace of Rd, by the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace, while ϕ(U)HJ,ϕ(u)(Rd) as the domain of πJ|ϕ(U)HJ,ϕ(u)(Rd) is the subspace of HJ,ϕ(u)(Rd), but as HJ,ϕ(u)(Rd) is the subspace of Rd, the domain is the subspace of Rd, likewise.

Let us suppose that the chart is a boundary chart.

ϕ(U) is an open subset of Hd and ϕ(U)HJ,ϕ(u)(Rd)=ϕ(U)HJ,ϕ(u)(Rd)Hd.

So, ϕ(HJ,u(U)) is an open subset of HJ,ϕ(u)(Rd)HdHd.

So, πJ(ϕ(HJ,u(U)))=πJ|HJ,ϕ(u)(Rd)Hd(ϕ(HJ,u(U))) is an open subset of Hd.

πJϕ|HJ,u(U):HJ,u(U)UπJ(ϕ(HJ,u(U)))Hd is a homeomorphism, because it is πJ|ϕ(U)HJ,ϕ(u)(Rd)Hdϕ|HJ,u(U), and ϕ|HJ,u(U):HJ,u(U)Uϕ(U)HJ,ϕ(u)(Rd)Hdϕ(U) and πJ|ϕ(U)HJ,ϕ(u)(Rd)Hd:ϕ(U)HJ,ϕ(u)(Rd)HdHJ,ϕ(u)(Rd)HdπJ(ϕ(HJ,u(U)))Hd are some homeomorphisms as some restrictions of homeomorphic ϕ and πJ|HJ,ϕ(u)(Rd)Hd: ϕ(U)HJ,ϕ(u)(Rd)Hd as the codomain of ϕ|HJ,u(U) is the subspace of ϕ(U), but as ϕ(U) is the subspace of Hd, the codomain is the subspace of Hd, as before, while ϕ(U)HJ,ϕ(u)(Rd)Hd as the domain of πJ|ϕ(U)HJ,ϕ(u)(Rd)Hd is the subspace of HJ,ϕ(u)(Rd)Hd, but as HJ,ϕ(u)(Rd)Hd is the subspace of Hd, the domain is the subspace of Hd, likewise.

The purpose of requiring ϕ(u)jd=0 is only that HJ,u(U) has at least 1 boundary point.

It is not that HJ,u(U) would be critically bad if it had no boundary point, but it would be useless for our purpose, because we are considering HJ,u(U) in order to allow a boundary point.

But is that requirement not too restrictive? No, we are considering 'J-half-slice of chart domain with respect to point' in order to have some boundary points (otherwise, 'J-slice of chart domain with respect to point' would be enough), and we can just choose u as one of them and if ϕ(u)jd0, we can just translate ϕ to make ϕ(u)jd=0.

The reason why we have elaborated on those facts is that we are going to construct a chart, (HJ,u(U)S,πJϕ|HJ,u(U)), for any SM that satisfies a certain condition (called "local-slice-or-half-slice condition"), to make S an embedded submanifold with boundary of M.


References


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