Mixing
Is the condition 'connected' necessary for differentiable structure?
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I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness condition is included mistakenly.
1.3 Definitions $;$ A locally Euclidean space $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $mathbb R^d$. If $varphi$ is a homeomorphism of a connected open set $Usubset M$ onto an open subset of $mathbb R^d$, $varphi$ is called a coordinate map, the functions $x_i = r_i circ varphi$ are called the coordinate functions, and the pair $(U, varphi)$ (somethimes denoted by $(U, x_1, ldots, x_d)$) is called a coordinate system. (The rest omitted because it is irrelavent to my question.)
1.4 Definitions $;$ A differentiable structure $mathcal F$ of class $C^k$ $(1leq kleq infty)$ on a locally Euclidean space $M$ is a collection of coordinate systems ${(U_alpha, varphi_alpha): alphain A}$ satisfying the following three properties:
(a) $bigcup_{alphain A}U_alpha = M$.
(b) $varphi_alphacircvarphi_beta^{-1}$ is $C^k$ for all $alpha,betain A$.
(c) The collection $mathcal F$ is maximal with respect to (b); that is, if $(U,varphi)$ is a coordinate system such that $varphicircvarphi_alpha^{-1}$ and $varphi_alphacircvarphi^{-1}$ are $C^k$ for all $alphain A$, then $(U,varphi)inmathcal F$.
And here is the definition of differentiable manifold in the book.
A $d$-dimensional differentiable manifold of class $C^k$ is a pair $(M, mathcal F)$ consisting of a $d$-dimensional, second countable, locally Euclidean space $M$ together with a differentiable structure $mathcal F$ of class $C^k$.
My question is: Is it standard to put connectedness condition in definition 1.3? Becuase the book seems to ignore the connectedness condition in the texts that follows the above definitions (I'm not sure though, because I only read very little). For example, see the following.
An open subset $U$ of a differentiable manifold $(M,mathcal F_M)$ is itself a differentiable manifold with the differentiable structure $mathcal F_U = {(U_alphacap U,varphi_alpha|U_alphacap U):(U_alpha,varphi_alpha)inmathcal F_M)$.
Isn't this incorrect if there is a connectedness condition in definition 1.3?
smooth-manifolds
asked Jan 7 at 5:29
zxcvzxcv
1269
$endgroup$
|
show 1 more comment
$begingroup$
I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness condition is included mistakenly.
1.3 Definitions $;$ A locally Euclidean space $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $mathbb R^d$. If $varphi$ is a homeomorphism of a connected open set $Usubset M$ onto an open subset of $mathbb R^d$, $varphi$ is called a coordinate map, the functions $x_i = r_i circ varphi$ are called the coordinate functions, and the pair $(U, varphi)$ (somethimes denoted by $(U, x_1, ldots, x_d)$) is called a coordinate system. (The rest omitted because it is irrelavent to my question.)
1.4 Definitions $;$ A differentiable structure $mathcal F$ of class $C^k$ $(1leq kleq infty)$ on a locally Euclidean space $M$ is a collection of coordinate systems ${(U_alpha, varphi_alpha): alphain A}$ satisfying the following three properties:
(a) $bigcup_{alphain A}U_alpha = M$.
(b) $varphi_alphacircvarphi_beta^{-1}$ is $C^k$ for all $alpha,betain A$.
(c) The collection $mathcal F$ is maximal with respect to (b); that is, if $(U,varphi)$ is a coordinate system such that $varphicircvarphi_alpha^{-1}$ and $varphi_alphacircvarphi^{-1}$ are $C^k$ for all $alphain A$, then $(U,varphi)inmathcal F$.
And here is the definition of differentiable manifold in the book.
A $d$-dimensional differentiable manifold of class $C^k$ is a pair $(M, mathcal F)$ consisting of a $d$-dimensional, second countable, locally Euclidean space $M$ together with a differentiable structure $mathcal F$ of class $C^k$.
My question is: Is it standard to put connectedness condition in definition 1.3? Becuase the book seems to ignore the connectedness condition in the texts that follows the above definitions (I'm not sure though, because I only read very little). For example, see the following.
An open subset $U$ of a differentiable manifold $(M,mathcal F_M)$ is itself a differentiable manifold with the differentiable structure $mathcal F_U = {(U_alphacap U,varphi_alpha|U_alphacap U):(U_alpha,varphi_alpha)inmathcal F_M)$.
Isn't this incorrect if there is a connectedness condition in definition 1.3?
smooth-manifolds
asked Jan 7 at 5:29
zxcvzxcv
1269
$endgroup$
$begingroup$
The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 5:34
$begingroup$
More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts?
$endgroup$
– stressed out
Jan 7 at 6:15
$begingroup$
@Lord Why is it redundant? If there were no connectedness condition, can't $(mathbb R, mathcal F)$ be a differentiable manifold, where $mathcal F$ is a differentiable structure containing $varphi_1: mathbb R-{0}tomathbb R-{0}, xmapsto x$ and $varphi_2: mathbb Rtomathbb R, xmapsto x$? Having a connectedness condition will not allow $varphi_1$ to be in $mathcal F$
$endgroup$
– zxcv
Jan 7 at 7:37
$begingroup$
@stressed Isn't it restrictive, as in my above comment?
$endgroup$
– zxcv
Jan 7 at 7:39
$begingroup$
@zxcv $Bbb R$ is a differentiable manifold.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 7:46
|
show 1 more comment
$begingroup$
I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness condition is included mistakenly.
1.3 Definitions $;$ A locally Euclidean space $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $mathbb R^d$. If $varphi$ is a homeomorphism of a connected open set $Usubset M$ onto an open subset of $mathbb R^d$, $varphi$ is called a coordinate map, the functions $x_i = r_i circ varphi$ are called the coordinate functions, and the pair $(U, varphi)$ (somethimes denoted by $(U, x_1, ldots, x_d)$) is called a coordinate system. (The rest omitted because it is irrelavent to my question.)
1.4 Definitions $;$ A differentiable structure $mathcal F$ of class $C^k$ $(1leq kleq infty)$ on a locally Euclidean space $M$ is a collection of coordinate systems ${(U_alpha, varphi_alpha): alphain A}$ satisfying the following three properties:
(a) $bigcup_{alphain A}U_alpha = M$.
(b) $varphi_alphacircvarphi_beta^{-1}$ is $C^k$ for all $alpha,betain A$.
(c) The collection $mathcal F$ is maximal with respect to (b); that is, if $(U,varphi)$ is a coordinate system such that $varphicircvarphi_alpha^{-1}$ and $varphi_alphacircvarphi^{-1}$ are $C^k$ for all $alphain A$, then $(U,varphi)inmathcal F$.
And here is the definition of differentiable manifold in the book.
A $d$-dimensional differentiable manifold of class $C^k$ is a pair $(M, mathcal F)$ consisting of a $d$-dimensional, second countable, locally Euclidean space $M$ together with a differentiable structure $mathcal F$ of class $C^k$.
My question is: Is it standard to put connectedness condition in definition 1.3? Becuase the book seems to ignore the connectedness condition in the texts that follows the above definitions (I'm not sure though, because I only read very little). For example, see the following.
An open subset $U$ of a differentiable manifold $(M,mathcal F_M)$ is itself a differentiable manifold with the differentiable structure $mathcal F_U = {(U_alphacap U,varphi_alpha|U_alphacap U):(U_alpha,varphi_alpha)inmathcal F_M)$.
Isn't this incorrect if there is a connectedness condition in definition 1.3?
smooth-manifolds
asked Jan 7 at 5:29
zxcvzxcv
1269
$endgroup$
I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness condition is included mistakenly.
1.3 Definitions $;$ A locally Euclidean space $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $mathbb R^d$. If $varphi$ is a homeomorphism of a connected open set $Usubset M$ onto an open subset of $mathbb R^d$, $varphi$ is called a coordinate map, the functions $x_i = r_i circ varphi$ are called the coordinate functions, and the pair $(U, varphi)$ (somethimes denoted by $(U, x_1, ldots, x_d)$) is called a coordinate system. (The rest omitted because it is irrelavent to my question.)
1.4 Definitions $;$ A differentiable structure $mathcal F$ of class $C^k$ $(1leq kleq infty)$ on a locally Euclidean space $M$ is a collection of coordinate systems ${(U_alpha, varphi_alpha): alphain A}$ satisfying the following three properties:
(a) $bigcup_{alphain A}U_alpha = M$.
(b) $varphi_alphacircvarphi_beta^{-1}$ is $C^k$ for all $alpha,betain A$.
(c) The collection $mathcal F$ is maximal with respect to (b); that is, if $(U,varphi)$ is a coordinate system such that $varphicircvarphi_alpha^{-1}$ and $varphi_alphacircvarphi^{-1}$ are $C^k$ for all $alphain A$, then $(U,varphi)inmathcal F$.
And here is the definition of differentiable manifold in the book.
A $d$-dimensional differentiable manifold of class $C^k$ is a pair $(M, mathcal F)$ consisting of a $d$-dimensional, second countable, locally Euclidean space $M$ together with a differentiable structure $mathcal F$ of class $C^k$.
My question is: Is it standard to put connectedness condition in definition 1.3? Becuase the book seems to ignore the connectedness condition in the texts that follows the above definitions (I'm not sure though, because I only read very little). For example, see the following.
An open subset $U$ of a differentiable manifold $(M,mathcal F_M)$ is itself a differentiable manifold with the differentiable structure $mathcal F_U = {(U_alphacap U,varphi_alpha|U_alphacap U):(U_alpha,varphi_alpha)inmathcal F_M)$.
Isn't this incorrect if there is a connectedness condition in definition 1.3?
smooth-manifolds
smooth-manifolds
asked Jan 7 at 5:29
zxcvzxcv
1269
asked Jan 7 at 5:29
zxcvzxcv
1269
edited Jan 7 at 6:39
zxcv
asked Jan 7 at 5:29
zxcvzxcv
1269
asked Jan 7 at 5:29
zxcvzxcv
1269
asked Jan 7 at 5:29
zxcvzxcv
1269
1269
$begingroup$
The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 5:34
$begingroup$
More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts?
$endgroup$
– stressed out
Jan 7 at 6:15
$begingroup$
@Lord Why is it redundant? If there were no connectedness condition, can't $(mathbb R, mathcal F)$ be a differentiable manifold, where $mathcal F$ is a differentiable structure containing $varphi_1: mathbb R-{0}tomathbb R-{0}, xmapsto x$ and $varphi_2: mathbb Rtomathbb R, xmapsto x$? Having a connectedness condition will not allow $varphi_1$ to be in $mathcal F$
$endgroup$
– zxcv
Jan 7 at 7:37
$begingroup$
@stressed Isn't it restrictive, as in my above comment?
$endgroup$
– zxcv
Jan 7 at 7:39
$begingroup$
@zxcv $Bbb R$ is a differentiable manifold.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 7:46
|
show 1 more comment
$begingroup$
The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 5:34
$begingroup$
More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts?
$endgroup$
– stressed out
Jan 7 at 6:15
$begingroup$
@Lord Why is it redundant? If there were no connectedness condition, can't $(mathbb R, mathcal F)$ be a differentiable manifold, where $mathcal F$ is a differentiable structure containing $varphi_1: mathbb R-{0}tomathbb R-{0}, xmapsto x$ and $varphi_2: mathbb Rtomathbb R, xmapsto x$? Having a connectedness condition will not allow $varphi_1$ to be in $mathcal F$
$endgroup$
– zxcv
Jan 7 at 7:37
$begingroup$
@stressed Isn't it restrictive, as in my above comment?
$endgroup$
– zxcv
Jan 7 at 7:39
$begingroup$
@zxcv $Bbb R$ is a differentiable manifold.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 7:46
$begingroup$
The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 5:34
$begingroup$
The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 5:34
$begingroup$
More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts?
$endgroup$
– stressed out
Jan 7 at 6:15
$begingroup$
More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts?
$endgroup$
– stressed out
Jan 7 at 6:15
$begingroup$
@Lord Why is it redundant? If there were no connectedness condition, can't $(mathbb R, mathcal F)$ be a differentiable manifold, where $mathcal F$ is a differentiable structure containing $varphi_1: mathbb R-{0}tomathbb R-{0}, xmapsto x$ and $varphi_2: mathbb Rtomathbb R, xmapsto x$? Having a connectedness condition will not allow $varphi_1$ to be in $mathcal F$
$endgroup$
– zxcv
Jan 7 at 7:37
$begingroup$
@Lord Why is it redundant? If there were no connectedness condition, can't $(mathbb R, mathcal F)$ be a differentiable manifold, where $mathcal F$ is a differentiable structure containing $varphi_1: mathbb R-{0}tomathbb R-{0}, xmapsto x$ and $varphi_2: mathbb Rtomathbb R, xmapsto x$? Having a connectedness condition will not allow $varphi_1$ to be in $mathcal F$
$endgroup$
– zxcv
Jan 7 at 7:37
$begingroup$
@stressed Isn't it restrictive, as in my above comment?
$endgroup$
– zxcv
Jan 7 at 7:39
$begingroup$
@stressed Isn't it restrictive, as in my above comment?
$endgroup$
– zxcv
Jan 7 at 7:39
$begingroup$
@zxcv $Bbb R$ is a differentiable manifold.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 7:46
$begingroup$
@zxcv $Bbb R$ is a differentiable manifold.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 7:46
|
show 1 more comment
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$begingroup$
The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 5:34
$begingroup$
More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts?
$endgroup$
– stressed out
Jan 7 at 6:15
$begingroup$
@Lord Why is it redundant? If there were no connectedness condition, can't $(mathbb R, mathcal F)$ be a differentiable manifold, where $mathcal F$ is a differentiable structure containing $varphi_1: mathbb R-{0}tomathbb R-{0}, xmapsto x$ and $varphi_2: mathbb Rtomathbb R, xmapsto x$? Having a connectedness condition will not allow $varphi_1$ to be in $mathcal F$
$endgroup$
– zxcv
Jan 7 at 7:37
$begingroup$
@stressed Isn't it restrictive, as in my above comment?
$endgroup$
– zxcv
Jan 7 at 7:39
$begingroup$
@zxcv $Bbb R$ is a differentiable manifold.
$endgroup$
– Lord Shark the Unknown
Jan 7 at 7:46