Where is this Definition of a Manifold from?

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Let $Msubset mathbbR^N$. We say that $M$ is a smooth submanifold of codimension $d$ or dimension $N-d=n$, if for every $pin M$ there is a neighbourhood $U$ of $p$ and smooth functions $f_1,...,f_d$ on $U$ which have the properties:



  1. $mathrmdf_1,mathrmdf_2,...,mathrmdf_d$ are linearly independent in $T^*mathbbR^N$

  2. $Ucap M = f_1=...=f_d=0 $

I have never seen this definition of a manifold and would like to know if there are any books working with it. I think this definition is closely related to the regular value theorem which is, again dependend on the exact definition of a manifold, a follow up theorem.




differential-geometry manifolds differential-topology smooth-manifolds




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    Probably all introductions to differential geometry prove, or leave it as an exercise, that that is an equivalent definition of an embedded manifold, using the implicit function theorem, or the inverse function theorem.
    – user582578
    Aug 8 at 16:18















up vote
1
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Let $Msubset mathbbR^N$. We say that $M$ is a smooth submanifold of codimension $d$ or dimension $N-d=n$, if for every $pin M$ there is a neighbourhood $U$ of $p$ and smooth functions $f_1,...,f_d$ on $U$ which have the properties:



  1. $mathrmdf_1,mathrmdf_2,...,mathrmdf_d$ are linearly independent in $T^*mathbbR^N$

  2. $Ucap M = f_1=...=f_d=0 $

I have never seen this definition of a manifold and would like to know if there are any books working with it. I think this definition is closely related to the regular value theorem which is, again dependend on the exact definition of a manifold, a follow up theorem.




differential-geometry manifolds differential-topology smooth-manifolds




share|cite|improve this question


















  • 1




    Probably all introductions to differential geometry prove, or leave it as an exercise, that that is an equivalent definition of an embedded manifold, using the implicit function theorem, or the inverse function theorem.
    – user582578
    Aug 8 at 16:18













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let $Msubset mathbbR^N$. We say that $M$ is a smooth submanifold of codimension $d$ or dimension $N-d=n$, if for every $pin M$ there is a neighbourhood $U$ of $p$ and smooth functions $f_1,...,f_d$ on $U$ which have the properties:



  1. $mathrmdf_1,mathrmdf_2,...,mathrmdf_d$ are linearly independent in $T^*mathbbR^N$

  2. $Ucap M = f_1=...=f_d=0 $

I have never seen this definition of a manifold and would like to know if there are any books working with it. I think this definition is closely related to the regular value theorem which is, again dependend on the exact definition of a manifold, a follow up theorem.




differential-geometry manifolds differential-topology smooth-manifolds




share|cite|improve this question














Let $Msubset mathbbR^N$. We say that $M$ is a smooth submanifold of codimension $d$ or dimension $N-d=n$, if for every $pin M$ there is a neighbourhood $U$ of $p$ and smooth functions $f_1,...,f_d$ on $U$ which have the properties:



  1. $mathrmdf_1,mathrmdf_2,...,mathrmdf_d$ are linearly independent in $T^*mathbbR^N$

  2. $Ucap M = f_1=...=f_d=0 $

I have never seen this definition of a manifold and would like to know if there are any books working with it. I think this definition is closely related to the regular value theorem which is, again dependend on the exact definition of a manifold, a follow up theorem.





differential-geometry manifolds differential-topology smooth-manifolds





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edited Aug 8 at 16:03









José Carlos Santos

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asked Aug 8 at 16:01









EpsilonDelta

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  • 1




    Probably all introductions to differential geometry prove, or leave it as an exercise, that that is an equivalent definition of an embedded manifold, using the implicit function theorem, or the inverse function theorem.
    – user582578
    Aug 8 at 16:18













  • 1




    Probably all introductions to differential geometry prove, or leave it as an exercise, that that is an equivalent definition of an embedded manifold, using the implicit function theorem, or the inverse function theorem.
    – user582578
    Aug 8 at 16:18








1




1




Probably all introductions to differential geometry prove, or leave it as an exercise, that that is an equivalent definition of an embedded manifold, using the implicit function theorem, or the inverse function theorem.
– user582578
Aug 8 at 16:18





Probably all introductions to differential geometry prove, or leave it as an exercise, that that is an equivalent definition of an embedded manifold, using the implicit function theorem, or the inverse function theorem.
– user582578
Aug 8 at 16:18











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If I remember well, there is a similar definition in Berger /Gostiaux book Differential Geometry: manifolds, curves and surfaces..



I have (unfortunately not at hand) a paper copy of the original French book.



The idea is similar to the way to define a linear subspace using linear forms.






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  • You are right, Definition 2.1.1 is like that in that book. See screenshot.
    – user582578
    Aug 8 at 16:23











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










If I remember well, there is a similar definition in Berger /Gostiaux book Differential Geometry: manifolds, curves and surfaces..



I have (unfortunately not at hand) a paper copy of the original French book.



The idea is similar to the way to define a linear subspace using linear forms.






share|cite|improve this answer




















  • You are right, Definition 2.1.1 is like that in that book. See screenshot.
    – user582578
    Aug 8 at 16:23















up vote
1
down vote



accepted










If I remember well, there is a similar definition in Berger /Gostiaux book Differential Geometry: manifolds, curves and surfaces..



I have (unfortunately not at hand) a paper copy of the original French book.



The idea is similar to the way to define a linear subspace using linear forms.






share|cite|improve this answer




















  • You are right, Definition 2.1.1 is like that in that book. See screenshot.
    – user582578
    Aug 8 at 16:23













up vote
1
down vote



accepted







up vote
1
down vote



accepted






If I remember well, there is a similar definition in Berger /Gostiaux book Differential Geometry: manifolds, curves and surfaces..



I have (unfortunately not at hand) a paper copy of the original French book.



The idea is similar to the way to define a linear subspace using linear forms.






share|cite|improve this answer












If I remember well, there is a similar definition in Berger /Gostiaux book Differential Geometry: manifolds, curves and surfaces..



I have (unfortunately not at hand) a paper copy of the original French book.



The idea is similar to the way to define a linear subspace using linear forms.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Aug 8 at 16:19









mathcounterexamples.net

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  • You are right, Definition 2.1.1 is like that in that book. See screenshot.
    – user582578
    Aug 8 at 16:23

















  • You are right, Definition 2.1.1 is like that in that book. See screenshot.
    – user582578
    Aug 8 at 16:23
















You are right, Definition 2.1.1 is like that in that book. See screenshot.
– user582578
Aug 8 at 16:23





You are right, Definition 2.1.1 is like that in that book. See screenshot.
– user582578
Aug 8 at 16:23













 

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