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Do Carmo Riemannian Geometry, chapter 0. Definition 2.1.

1

$begingroup$

A differentiable manifold of dimension $n$ is a set $M$ and a family of injective mappings $x_alpha : U_alpha subset mathbbR^n to M$ of open sets $U_alpha$ of $mathbbR^n$ such that:

1) $bigcup_alpha U_alpha = M$.

2) For any pair $alpha,beta$, with $x_alpha(U_alpha) cap x_beta (U_beta) = W neq emptyset$, the sets $x^-1_alpha(W)$ and $x^-1_beta(W)$ are open sets in $mathbbR^n$ and the mappings $x_beta^-1 circ x_alpha$ are differentiable.

3) The family $left(U_alpha,x_alpha )right$ is maximal relative to condition 1) and 2).

What does maximality mean in this context?

differential-geometry definition

share|cite|improve this question

edited Apr 3 at 19:04

Ernie060

2,950719

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @user647486: That's not quite correct. For instance, on $mathbf R$, there are two distinct smooth structures contained in different maximal atlases, one generated by the global chart $(mathbf R, mathrmId)$ and another generated by $(mathbf R,xmapsto x^3)$. These are not compatible smooth structures since $xmapsto x^1/3$ is not differentiable everywhere it is defined.
  $endgroup$
  – Alex Ortiz
  Apr 2 at 17:00
  

  
  
  
  

add a comment | 

1

$begingroup$

A differentiable manifold of dimension $n$ is a set $M$ and a family of injective mappings $x_alpha : U_alpha subset mathbbR^n to M$ of open sets $U_alpha$ of $mathbbR^n$ such that:

1) $bigcup_alpha U_alpha = M$.

2) For any pair $alpha,beta$, with $x_alpha(U_alpha) cap x_beta (U_beta) = W neq emptyset$, the sets $x^-1_alpha(W)$ and $x^-1_beta(W)$ are open sets in $mathbbR^n$ and the mappings $x_beta^-1 circ x_alpha$ are differentiable.

3) The family $left(U_alpha,x_alpha )right$ is maximal relative to condition 1) and 2).

What does maximality mean in this context?

differential-geometry definition

share|cite|improve this question

edited Apr 3 at 19:04

Ernie060

2,950719

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @user647486: That's not quite correct. For instance, on $mathbf R$, there are two distinct smooth structures contained in different maximal atlases, one generated by the global chart $(mathbf R, mathrmId)$ and another generated by $(mathbf R,xmapsto x^3)$. These are not compatible smooth structures since $xmapsto x^1/3$ is not differentiable everywhere it is defined.
  $endgroup$
  – Alex Ortiz
  Apr 2 at 17:00
  

  
  
  
  

add a comment | 

$begingroup$

A differentiable manifold of dimension $n$ is a set $M$ and a family of injective mappings $x_alpha : U_alpha subset mathbbR^n to M$ of open sets $U_alpha$ of $mathbbR^n$ such that:

1) $bigcup_alpha U_alpha = M$.

2) For any pair $alpha,beta$, with $x_alpha(U_alpha) cap x_beta (U_beta) = W neq emptyset$, the sets $x^-1_alpha(W)$ and $x^-1_beta(W)$ are open sets in $mathbbR^n$ and the mappings $x_beta^-1 circ x_alpha$ are differentiable.

3) The family $left(U_alpha,x_alpha )right$ is maximal relative to condition 1) and 2).

What does maximality mean in this context?

differential-geometry definition

share|cite|improve this question

edited Apr 3 at 19:04

Ernie060

2,950719

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

$endgroup$

share|cite|improve this question

edited Apr 3 at 19:04

Ernie060

2,950719

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

share|cite|improve this question

edited Apr 3 at 19:04

Ernie060

2,950719

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

edited Apr 3 at 19:04

Ernie060

2,950719

edited Apr 3 at 19:04

Ernie060

2,950719

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

asked Apr 2 at 16:53

user8469759user8469759

1,5731618

1 Answer
1

active

oldest

votes

3

$begingroup$

Maximality in this context means that if $(V_beta,y_beta)$ is another collection of injective mappings $y_beta$ and open sets $V_beta$ such that $bigcup_beta V_beta = M$, and such that for each $alpha$ and each $beta$, the mappings $x_alphacirc y_beta^-1$ and $y_betacirc x_alpha^-1$ are differentiable where they are defined, then $(V_beta,y_beta)subset(U_alpha,x_alpha)$.

Another way of stating maximality is to say that the collection $(U_alpha,x_alpha)$ contains every chart $(V,y)$ that is compatible with $(U_alpha,x_alpha)$ for each $alpha$.

share|cite|improve this answer

edited Apr 2 at 17:12

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442

$endgroup$

add a comment | 

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3

$begingroup$

Maximality in this context means that if $(V_beta,y_beta)$ is another collection of injective mappings $y_beta$ and open sets $V_beta$ such that $bigcup_beta V_beta = M$, and such that for each $alpha$ and each $beta$, the mappings $x_alphacirc y_beta^-1$ and $y_betacirc x_alpha^-1$ are differentiable where they are defined, then $(V_beta,y_beta)subset(U_alpha,x_alpha)$.

Another way of stating maximality is to say that the collection $(U_alpha,x_alpha)$ contains every chart $(V,y)$ that is compatible with $(U_alpha,x_alpha)$ for each $alpha$.

share|cite|improve this answer

edited Apr 2 at 17:12

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442

$endgroup$

add a comment | 

3

$begingroup$

Maximality in this context means that if $(V_beta,y_beta)$ is another collection of injective mappings $y_beta$ and open sets $V_beta$ such that $bigcup_beta V_beta = M$, and such that for each $alpha$ and each $beta$, the mappings $x_alphacirc y_beta^-1$ and $y_betacirc x_alpha^-1$ are differentiable where they are defined, then $(V_beta,y_beta)subset(U_alpha,x_alpha)$.

Another way of stating maximality is to say that the collection $(U_alpha,x_alpha)$ contains every chart $(V,y)$ that is compatible with $(U_alpha,x_alpha)$ for each $alpha$.

share|cite|improve this answer

edited Apr 2 at 17:12

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442

$endgroup$

add a comment | 

$begingroup$

Maximality in this context means that if $(V_beta,y_beta)$ is another collection of injective mappings $y_beta$ and open sets $V_beta$ such that $bigcup_beta V_beta = M$, and such that for each $alpha$ and each $beta$, the mappings $x_alphacirc y_beta^-1$ and $y_betacirc x_alpha^-1$ are differentiable where they are defined, then $(V_beta,y_beta)subset(U_alpha,x_alpha)$.

Another way of stating maximality is to say that the collection $(U_alpha,x_alpha)$ contains every chart $(V,y)$ that is compatible with $(U_alpha,x_alpha)$ for each $alpha$.

share|cite|improve this answer

edited Apr 2 at 17:12

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442

$endgroup$

share|cite|improve this answer

edited Apr 2 at 17:12

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442

answered Apr 2 at 16:58

Alex OrtizAlex Ortiz

11.6k21442