- #1
- 7,802
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Hi, everyone:
I have been reading Boothby's intro to diff. manifolds, and in def. 4.3,
talking about 1-1 immersions F:N->M , n and m-mfld. and M an m-mfld.
That:
"...F establishes a 1-1 correspondence between N and the image N'=F(N)
of M. If we use this correspondence to give N' a topology and a C^oo
structure, then N' will be called a submanifold, and F:N->N' a diffeomorphism".
I am just wondering how this topology and C^oo structure are defined.
I imagine we are using pullbacks here, tho, since F is not necessarily
a diffeom. , we cannot pullback by F, except maybe locally, using
the inverse fn. theorem.
And, re the topology on F'(N) , we cannot use subspace, since
F may not be an embedding. Maybe we are using quotient topology,
but it does not seem clear.
Anyone know how the topology and C^oo structure are defined
in N'=F(N)?
I would appreciate any help.
I have been reading Boothby's intro to diff. manifolds, and in def. 4.3,
talking about 1-1 immersions F:N->M , n and m-mfld. and M an m-mfld.
That:
"...F establishes a 1-1 correspondence between N and the image N'=F(N)
of M. If we use this correspondence to give N' a topology and a C^oo
structure, then N' will be called a submanifold, and F:N->N' a diffeomorphism".
I am just wondering how this topology and C^oo structure are defined.
I imagine we are using pullbacks here, tho, since F is not necessarily
a diffeom. , we cannot pullback by F, except maybe locally, using
the inverse fn. theorem.
And, re the topology on F'(N) , we cannot use subspace, since
F may not be an embedding. Maybe we are using quotient topology,
but it does not seem clear.
Anyone know how the topology and C^oo structure are defined
in N'=F(N)?
I would appreciate any help.