Mixing
What is a differetial structure, exactly?
$begingroup$
A structure in general is a set and some operations on that set or ordersrelations of some kind.
In algebra and topology this is rather clear, but in differential geometry one often consider "differential structure".
I understand that an atlas is related to this matter and my impression is that an atlas induces a differential structure. But what is this structure?
In topology we can relate the structure to continuity i.e any topological space space that has the same topological structure also have the same continuous functions(even if it is not defined on the exact same sets). Hence one would think that differential structure and differentiable functions have the same dynamics.
Something tells me this is related to the tangent spaces and how they look as linear spaces since this is what the atlas induces at each point via the partials of the charts.
Does anyone have good answer for what the differential structure consists of or how to think about it? Two different manifolds with different atlases should be able to have the same "differential structure" as far as I understand.
differential-geometry riemannian-geometry
asked Jan 31 at 5:18
MaxedMaxed
2411527
$endgroup$
|
show 3 more comments
$begingroup$
A structure in general is a set and some operations on that set or ordersrelations of some kind.
In algebra and topology this is rather clear, but in differential geometry one often consider "differential structure".
I understand that an atlas is related to this matter and my impression is that an atlas induces a differential structure. But what is this structure?
In topology we can relate the structure to continuity i.e any topological space space that has the same topological structure also have the same continuous functions(even if it is not defined on the exact same sets). Hence one would think that differential structure and differentiable functions have the same dynamics.
Something tells me this is related to the tangent spaces and how they look as linear spaces since this is what the atlas induces at each point via the partials of the charts.
Does anyone have good answer for what the differential structure consists of or how to think about it? Two different manifolds with different atlases should be able to have the same "differential structure" as far as I understand.
differential-geometry riemannian-geometry
asked Jan 31 at 5:18
MaxedMaxed
2411527
$endgroup$
2
$begingroup$
A differential structure is the same as an atlas, or more precisely a maximum atlas. It says what functions defined on the manifold are smooth in the same sense that a topology says what functions are continuous.
$endgroup$
– Dante Grevino
Jan 31 at 5:24
$begingroup$
@DanteGrevino hence it is wrong of me to consider it a more abstract notion? such an group? where two groups can be "algebraically equal" but still very different. A counterexample would be two different familes of charts(atlases) inducing the same differential structure. As long as the bases of the tanget spaces are kind of "similar" I suppose we would get similar differential structure, but that might not be the case.
$endgroup$
– Maxed
Jan 31 at 5:25
2
$begingroup$
Your description of the concept of a "structure" seems to have been imported from algebra. That's not the only concept of "structure" in mathematics. Indeed, mathematicians are willing to accept very different kinds of structure in different branches of mathematics.
$endgroup$
– Lee Mosher
Jan 31 at 15:10
3
$begingroup$
Just like two groups can be isomorphic without being equal, two differentialble manifolds can be diffeomorphic without being equal. The common ground here is the existence of a structure preserving map: a map of groups $f : G to K$ is an isomorphism if it is a bijection and it preserves the group structure (the binary operation and the inversion); a map $f : M to N$ of differentiable manifold is a difffeomorphism if it is a bijection and it preserves the differentiable structure (the pullback of an atlas of $N$ is an atlas of $M$).
$endgroup$
– Lee Mosher
Jan 31 at 15:12
$begingroup$
@LeeMosher I was afraid that might be the case.. stated here ; en.wikipedia.org/wiki/Mathematical_structure
$endgroup$
– Maxed
Jan 31 at 15:13
|
show 3 more comments
$begingroup$
A structure in general is a set and some operations on that set or ordersrelations of some kind.
In algebra and topology this is rather clear, but in differential geometry one often consider "differential structure".
I understand that an atlas is related to this matter and my impression is that an atlas induces a differential structure. But what is this structure?
In topology we can relate the structure to continuity i.e any topological space space that has the same topological structure also have the same continuous functions(even if it is not defined on the exact same sets). Hence one would think that differential structure and differentiable functions have the same dynamics.
Something tells me this is related to the tangent spaces and how they look as linear spaces since this is what the atlas induces at each point via the partials of the charts.
Does anyone have good answer for what the differential structure consists of or how to think about it? Two different manifolds with different atlases should be able to have the same "differential structure" as far as I understand.
differential-geometry riemannian-geometry
asked Jan 31 at 5:18
MaxedMaxed
2411527
$endgroup$
A structure in general is a set and some operations on that set or ordersrelations of some kind.
In algebra and topology this is rather clear, but in differential geometry one often consider "differential structure".
I understand that an atlas is related to this matter and my impression is that an atlas induces a differential structure. But what is this structure?
In topology we can relate the structure to continuity i.e any topological space space that has the same topological structure also have the same continuous functions(even if it is not defined on the exact same sets). Hence one would think that differential structure and differentiable functions have the same dynamics.
Something tells me this is related to the tangent spaces and how they look as linear spaces since this is what the atlas induces at each point via the partials of the charts.
Does anyone have good answer for what the differential structure consists of or how to think about it? Two different manifolds with different atlases should be able to have the same "differential structure" as far as I understand.
differential-geometry riemannian-geometry
differential-geometry riemannian-geometry
asked Jan 31 at 5:18
MaxedMaxed
2411527
asked Jan 31 at 5:18
MaxedMaxed
2411527
edited Jan 31 at 15:06
Maxed
asked Jan 31 at 5:18
MaxedMaxed
2411527
asked Jan 31 at 5:18
MaxedMaxed
2411527
asked Jan 31 at 5:18
MaxedMaxed
2411527
2411527
2
$begingroup$
A differential structure is the same as an atlas, or more precisely a maximum atlas. It says what functions defined on the manifold are smooth in the same sense that a topology says what functions are continuous.
$endgroup$
– Dante Grevino
Jan 31 at 5:24
$begingroup$
@DanteGrevino hence it is wrong of me to consider it a more abstract notion? such an group? where two groups can be "algebraically equal" but still very different. A counterexample would be two different familes of charts(atlases) inducing the same differential structure. As long as the bases of the tanget spaces are kind of "similar" I suppose we would get similar differential structure, but that might not be the case.
$endgroup$
– Maxed
Jan 31 at 5:25
2
$begingroup$
Your description of the concept of a "structure" seems to have been imported from algebra. That's not the only concept of "structure" in mathematics. Indeed, mathematicians are willing to accept very different kinds of structure in different branches of mathematics.
$endgroup$
– Lee Mosher
Jan 31 at 15:10
3
$begingroup$
Just like two groups can be isomorphic without being equal, two differentialble manifolds can be diffeomorphic without being equal. The common ground here is the existence of a structure preserving map: a map of groups $f : G to K$ is an isomorphism if it is a bijection and it preserves the group structure (the binary operation and the inversion); a map $f : M to N$ of differentiable manifold is a difffeomorphism if it is a bijection and it preserves the differentiable structure (the pullback of an atlas of $N$ is an atlas of $M$).
$endgroup$
– Lee Mosher
Jan 31 at 15:12
$begingroup$
@LeeMosher I was afraid that might be the case.. stated here ; en.wikipedia.org/wiki/Mathematical_structure
$endgroup$
– Maxed
Jan 31 at 15:13
|
show 3 more comments
2
$begingroup$
A differential structure is the same as an atlas, or more precisely a maximum atlas. It says what functions defined on the manifold are smooth in the same sense that a topology says what functions are continuous.
$endgroup$
– Dante Grevino
Jan 31 at 5:24
$begingroup$
@DanteGrevino hence it is wrong of me to consider it a more abstract notion? such an group? where two groups can be "algebraically equal" but still very different. A counterexample would be two different familes of charts(atlases) inducing the same differential structure. As long as the bases of the tanget spaces are kind of "similar" I suppose we would get similar differential structure, but that might not be the case.
$endgroup$
– Maxed
Jan 31 at 5:25
2
$begingroup$
Your description of the concept of a "structure" seems to have been imported from algebra. That's not the only concept of "structure" in mathematics. Indeed, mathematicians are willing to accept very different kinds of structure in different branches of mathematics.
$endgroup$
– Lee Mosher
Jan 31 at 15:10
3
$begingroup$
Just like two groups can be isomorphic without being equal, two differentialble manifolds can be diffeomorphic without being equal. The common ground here is the existence of a structure preserving map: a map of groups $f : G to K$ is an isomorphism if it is a bijection and it preserves the group structure (the binary operation and the inversion); a map $f : M to N$ of differentiable manifold is a difffeomorphism if it is a bijection and it preserves the differentiable structure (the pullback of an atlas of $N$ is an atlas of $M$).
$endgroup$
– Lee Mosher
Jan 31 at 15:12
$begingroup$
@LeeMosher I was afraid that might be the case.. stated here ; en.wikipedia.org/wiki/Mathematical_structure
$endgroup$
– Maxed
Jan 31 at 15:13
2
2
$begingroup$
A differential structure is the same as an atlas, or more precisely a maximum atlas. It says what functions defined on the manifold are smooth in the same sense that a topology says what functions are continuous.
$endgroup$
– Dante Grevino
Jan 31 at 5:24
$begingroup$
A differential structure is the same as an atlas, or more precisely a maximum atlas. It says what functions defined on the manifold are smooth in the same sense that a topology says what functions are continuous.
$endgroup$
– Dante Grevino
Jan 31 at 5:24
$begingroup$
@DanteGrevino hence it is wrong of me to consider it a more abstract notion? such an group? where two groups can be "algebraically equal" but still very different. A counterexample would be two different familes of charts(atlases) inducing the same differential structure. As long as the bases of the tanget spaces are kind of "similar" I suppose we would get similar differential structure, but that might not be the case.
$endgroup$
– Maxed
Jan 31 at 5:25
$begingroup$
@DanteGrevino hence it is wrong of me to consider it a more abstract notion? such an group? where two groups can be "algebraically equal" but still very different. A counterexample would be two different familes of charts(atlases) inducing the same differential structure. As long as the bases of the tanget spaces are kind of "similar" I suppose we would get similar differential structure, but that might not be the case.
$endgroup$
– Maxed
Jan 31 at 5:25
2
2
$begingroup$
Your description of the concept of a "structure" seems to have been imported from algebra. That's not the only concept of "structure" in mathematics. Indeed, mathematicians are willing to accept very different kinds of structure in different branches of mathematics.
$endgroup$
– Lee Mosher
Jan 31 at 15:10
$begingroup$
Your description of the concept of a "structure" seems to have been imported from algebra. That's not the only concept of "structure" in mathematics. Indeed, mathematicians are willing to accept very different kinds of structure in different branches of mathematics.
$endgroup$
– Lee Mosher
Jan 31 at 15:10
3
3
$begingroup$
Just like two groups can be isomorphic without being equal, two differentialble manifolds can be diffeomorphic without being equal. The common ground here is the existence of a structure preserving map: a map of groups $f : G to K$ is an isomorphism if it is a bijection and it preserves the group structure (the binary operation and the inversion); a map $f : M to N$ of differentiable manifold is a difffeomorphism if it is a bijection and it preserves the differentiable structure (the pullback of an atlas of $N$ is an atlas of $M$).
$endgroup$
– Lee Mosher
Jan 31 at 15:12
$begingroup$
Just like two groups can be isomorphic without being equal, two differentialble manifolds can be diffeomorphic without being equal. The common ground here is the existence of a structure preserving map: a map of groups $f : G to K$ is an isomorphism if it is a bijection and it preserves the group structure (the binary operation and the inversion); a map $f : M to N$ of differentiable manifold is a difffeomorphism if it is a bijection and it preserves the differentiable structure (the pullback of an atlas of $N$ is an atlas of $M$).
$endgroup$
– Lee Mosher
Jan 31 at 15:12
$begingroup$
@LeeMosher I was afraid that might be the case.. stated here ; en.wikipedia.org/wiki/Mathematical_structure
$endgroup$
– Maxed
Jan 31 at 15:13
$begingroup$
@LeeMosher I was afraid that might be the case.. stated here ; en.wikipedia.org/wiki/Mathematical_structure
$endgroup$
– Maxed
Jan 31 at 15:13
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Definitions:
Definition: Let $n$ be a non-negative integer number and let $M$ be a topological space. We say that $M$ is a topological $n$-manifold if it satisfies the following axioms.
1) $M$ is Hausdorff.
2) $M$ is second-countable, that is, there exists an at most countable basis for the topology.
3) $M$ is locally euclidean, that is, for every point $p$ in $M$ and every open neighborhood $U$ of $p$ there exists an homeomorphism $varphi : V to W$, where $V$ is an open neightborhood of $p$ contained in $U$ and $W$ is an open set of $mathbb{R}^n$.
In such case, a pair $(varphi,V)$, where $varphi:Vto W$ is an homeomorphism from an open set $V$ of $M$ to an open set $W$ of $mathbb{R}^n$, is called a chart of $M$.
Remark: For $n=0$, we agree $mathbb{R}^0={0}$ is a singleton endowed with the unique possible topology so a topological $0$-manifold is the same as an at most countable discrete space.
Definition: A smooth atlas, or more briefly an atlas, on a topological $n$-manifold $M$ is a set $mathcal{A} = {(varphi_j,V_j)mid jin J}$ of charts of $M$ that satisfy the following conditions.
1) The set ${V_jmid jin J}$ is an open cover of $M$, that is, $cup_{jin J}V_j=M$.
2) For every $i$ and $j$ in $J$ such that $V_icap V_jneqemptyset$ we have that $varphi_icirc varphi_j^{-1} : varphi_j(V_icap V_j)to varphi_i(V_icap V_j)$ is smooth, that is, infinitely differentiable.
Remark: Given a topological $n$-manifold $M$ such that there exists an smooth atlas on it, we can consider the poset (partial ordered set) of the set of atlas on $M$ ordered by inclusion.
Definition: A smooth $n$-manifold is a pair $(M,mathcal{A})$, where $M$ is a topological $n$-manifold and $mathcal{A}$ is a maximal smooth atlas on $M$.
Lemma: Let $M$ be a topological $n$-manifold. For every smooth atlas $mathcal{A}$ on $M$ there exists a unique maximal atlas $mathcal{A}^{text{max}}$ on $M$ containing $mathcal{A}$. Given two smooth atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ we have that $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$ if and only if $mathcal{A}_1cupmathcal{A}_2$ is an atlas.
For a proof see Proposition 1.17 in Lee.
Definition: A differential structure on a topological $n$-manifold $M$ is a maximal atlas $mathcal{A}$ on $M$. We say that two atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ determine the same differential structure if $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$.
Comments:
1) If you like to adopt an algebraic point of view, we could compare the properties of the poset of atlas on a topological manifold with the poset of topologies on a set and they are very different.
2) The atlas on manifolds say what functions between manifolds are smooth in the same sense that topologies say what functions between topological spaces are continuous. Smooth manifolds and smooth functions form a category and the same is true for topological spaces and continuous functions. In a category we have a notion of isomorphism which says that two objects are equivalent and, as @LeeMosher pointed out in the comments, in many cases it admits an interpretation in terms of preserving the structure. In the first case, an isomorphism is a diffeomorphism and, in the second case, it is an homeomorphism.
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
$endgroup$
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
1
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
2
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
|
show 1 more comment
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1 Answer
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1 Answer
1
active
oldest
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active
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$begingroup$
Definitions:
Definition: Let $n$ be a non-negative integer number and let $M$ be a topological space. We say that $M$ is a topological $n$-manifold if it satisfies the following axioms.
1) $M$ is Hausdorff.
2) $M$ is second-countable, that is, there exists an at most countable basis for the topology.
3) $M$ is locally euclidean, that is, for every point $p$ in $M$ and every open neighborhood $U$ of $p$ there exists an homeomorphism $varphi : V to W$, where $V$ is an open neightborhood of $p$ contained in $U$ and $W$ is an open set of $mathbb{R}^n$.
In such case, a pair $(varphi,V)$, where $varphi:Vto W$ is an homeomorphism from an open set $V$ of $M$ to an open set $W$ of $mathbb{R}^n$, is called a chart of $M$.
Remark: For $n=0$, we agree $mathbb{R}^0={0}$ is a singleton endowed with the unique possible topology so a topological $0$-manifold is the same as an at most countable discrete space.
Definition: A smooth atlas, or more briefly an atlas, on a topological $n$-manifold $M$ is a set $mathcal{A} = {(varphi_j,V_j)mid jin J}$ of charts of $M$ that satisfy the following conditions.
1) The set ${V_jmid jin J}$ is an open cover of $M$, that is, $cup_{jin J}V_j=M$.
2) For every $i$ and $j$ in $J$ such that $V_icap V_jneqemptyset$ we have that $varphi_icirc varphi_j^{-1} : varphi_j(V_icap V_j)to varphi_i(V_icap V_j)$ is smooth, that is, infinitely differentiable.
Remark: Given a topological $n$-manifold $M$ such that there exists an smooth atlas on it, we can consider the poset (partial ordered set) of the set of atlas on $M$ ordered by inclusion.
Definition: A smooth $n$-manifold is a pair $(M,mathcal{A})$, where $M$ is a topological $n$-manifold and $mathcal{A}$ is a maximal smooth atlas on $M$.
Lemma: Let $M$ be a topological $n$-manifold. For every smooth atlas $mathcal{A}$ on $M$ there exists a unique maximal atlas $mathcal{A}^{text{max}}$ on $M$ containing $mathcal{A}$. Given two smooth atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ we have that $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$ if and only if $mathcal{A}_1cupmathcal{A}_2$ is an atlas.
For a proof see Proposition 1.17 in Lee.
Definition: A differential structure on a topological $n$-manifold $M$ is a maximal atlas $mathcal{A}$ on $M$. We say that two atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ determine the same differential structure if $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$.
Comments:
1) If you like to adopt an algebraic point of view, we could compare the properties of the poset of atlas on a topological manifold with the poset of topologies on a set and they are very different.
2) The atlas on manifolds say what functions between manifolds are smooth in the same sense that topologies say what functions between topological spaces are continuous. Smooth manifolds and smooth functions form a category and the same is true for topological spaces and continuous functions. In a category we have a notion of isomorphism which says that two objects are equivalent and, as @LeeMosher pointed out in the comments, in many cases it admits an interpretation in terms of preserving the structure. In the first case, an isomorphism is a diffeomorphism and, in the second case, it is an homeomorphism.
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
$endgroup$
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
1
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
2
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
|
show 1 more comment
$begingroup$
Definitions:
Definition: Let $n$ be a non-negative integer number and let $M$ be a topological space. We say that $M$ is a topological $n$-manifold if it satisfies the following axioms.
1) $M$ is Hausdorff.
2) $M$ is second-countable, that is, there exists an at most countable basis for the topology.
3) $M$ is locally euclidean, that is, for every point $p$ in $M$ and every open neighborhood $U$ of $p$ there exists an homeomorphism $varphi : V to W$, where $V$ is an open neightborhood of $p$ contained in $U$ and $W$ is an open set of $mathbb{R}^n$.
In such case, a pair $(varphi,V)$, where $varphi:Vto W$ is an homeomorphism from an open set $V$ of $M$ to an open set $W$ of $mathbb{R}^n$, is called a chart of $M$.
Remark: For $n=0$, we agree $mathbb{R}^0={0}$ is a singleton endowed with the unique possible topology so a topological $0$-manifold is the same as an at most countable discrete space.
Definition: A smooth atlas, or more briefly an atlas, on a topological $n$-manifold $M$ is a set $mathcal{A} = {(varphi_j,V_j)mid jin J}$ of charts of $M$ that satisfy the following conditions.
1) The set ${V_jmid jin J}$ is an open cover of $M$, that is, $cup_{jin J}V_j=M$.
2) For every $i$ and $j$ in $J$ such that $V_icap V_jneqemptyset$ we have that $varphi_icirc varphi_j^{-1} : varphi_j(V_icap V_j)to varphi_i(V_icap V_j)$ is smooth, that is, infinitely differentiable.
Remark: Given a topological $n$-manifold $M$ such that there exists an smooth atlas on it, we can consider the poset (partial ordered set) of the set of atlas on $M$ ordered by inclusion.
Definition: A smooth $n$-manifold is a pair $(M,mathcal{A})$, where $M$ is a topological $n$-manifold and $mathcal{A}$ is a maximal smooth atlas on $M$.
Lemma: Let $M$ be a topological $n$-manifold. For every smooth atlas $mathcal{A}$ on $M$ there exists a unique maximal atlas $mathcal{A}^{text{max}}$ on $M$ containing $mathcal{A}$. Given two smooth atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ we have that $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$ if and only if $mathcal{A}_1cupmathcal{A}_2$ is an atlas.
For a proof see Proposition 1.17 in Lee.
Definition: A differential structure on a topological $n$-manifold $M$ is a maximal atlas $mathcal{A}$ on $M$. We say that two atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ determine the same differential structure if $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$.
Comments:
1) If you like to adopt an algebraic point of view, we could compare the properties of the poset of atlas on a topological manifold with the poset of topologies on a set and they are very different.
2) The atlas on manifolds say what functions between manifolds are smooth in the same sense that topologies say what functions between topological spaces are continuous. Smooth manifolds and smooth functions form a category and the same is true for topological spaces and continuous functions. In a category we have a notion of isomorphism which says that two objects are equivalent and, as @LeeMosher pointed out in the comments, in many cases it admits an interpretation in terms of preserving the structure. In the first case, an isomorphism is a diffeomorphism and, in the second case, it is an homeomorphism.
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
$endgroup$
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
1
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
2
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
|
show 1 more comment
$begingroup$
Definitions:
Definition: Let $n$ be a non-negative integer number and let $M$ be a topological space. We say that $M$ is a topological $n$-manifold if it satisfies the following axioms.
1) $M$ is Hausdorff.
2) $M$ is second-countable, that is, there exists an at most countable basis for the topology.
3) $M$ is locally euclidean, that is, for every point $p$ in $M$ and every open neighborhood $U$ of $p$ there exists an homeomorphism $varphi : V to W$, where $V$ is an open neightborhood of $p$ contained in $U$ and $W$ is an open set of $mathbb{R}^n$.
In such case, a pair $(varphi,V)$, where $varphi:Vto W$ is an homeomorphism from an open set $V$ of $M$ to an open set $W$ of $mathbb{R}^n$, is called a chart of $M$.
Remark: For $n=0$, we agree $mathbb{R}^0={0}$ is a singleton endowed with the unique possible topology so a topological $0$-manifold is the same as an at most countable discrete space.
Definition: A smooth atlas, or more briefly an atlas, on a topological $n$-manifold $M$ is a set $mathcal{A} = {(varphi_j,V_j)mid jin J}$ of charts of $M$ that satisfy the following conditions.
1) The set ${V_jmid jin J}$ is an open cover of $M$, that is, $cup_{jin J}V_j=M$.
2) For every $i$ and $j$ in $J$ such that $V_icap V_jneqemptyset$ we have that $varphi_icirc varphi_j^{-1} : varphi_j(V_icap V_j)to varphi_i(V_icap V_j)$ is smooth, that is, infinitely differentiable.
Remark: Given a topological $n$-manifold $M$ such that there exists an smooth atlas on it, we can consider the poset (partial ordered set) of the set of atlas on $M$ ordered by inclusion.
Definition: A smooth $n$-manifold is a pair $(M,mathcal{A})$, where $M$ is a topological $n$-manifold and $mathcal{A}$ is a maximal smooth atlas on $M$.
Lemma: Let $M$ be a topological $n$-manifold. For every smooth atlas $mathcal{A}$ on $M$ there exists a unique maximal atlas $mathcal{A}^{text{max}}$ on $M$ containing $mathcal{A}$. Given two smooth atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ we have that $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$ if and only if $mathcal{A}_1cupmathcal{A}_2$ is an atlas.
For a proof see Proposition 1.17 in Lee.
Definition: A differential structure on a topological $n$-manifold $M$ is a maximal atlas $mathcal{A}$ on $M$. We say that two atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ determine the same differential structure if $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$.
Comments:
1) If you like to adopt an algebraic point of view, we could compare the properties of the poset of atlas on a topological manifold with the poset of topologies on a set and they are very different.
2) The atlas on manifolds say what functions between manifolds are smooth in the same sense that topologies say what functions between topological spaces are continuous. Smooth manifolds and smooth functions form a category and the same is true for topological spaces and continuous functions. In a category we have a notion of isomorphism which says that two objects are equivalent and, as @LeeMosher pointed out in the comments, in many cases it admits an interpretation in terms of preserving the structure. In the first case, an isomorphism is a diffeomorphism and, in the second case, it is an homeomorphism.
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
$endgroup$
Definitions:
Definition: Let $n$ be a non-negative integer number and let $M$ be a topological space. We say that $M$ is a topological $n$-manifold if it satisfies the following axioms.
1) $M$ is Hausdorff.
2) $M$ is second-countable, that is, there exists an at most countable basis for the topology.
3) $M$ is locally euclidean, that is, for every point $p$ in $M$ and every open neighborhood $U$ of $p$ there exists an homeomorphism $varphi : V to W$, where $V$ is an open neightborhood of $p$ contained in $U$ and $W$ is an open set of $mathbb{R}^n$.
In such case, a pair $(varphi,V)$, where $varphi:Vto W$ is an homeomorphism from an open set $V$ of $M$ to an open set $W$ of $mathbb{R}^n$, is called a chart of $M$.
Remark: For $n=0$, we agree $mathbb{R}^0={0}$ is a singleton endowed with the unique possible topology so a topological $0$-manifold is the same as an at most countable discrete space.
Definition: A smooth atlas, or more briefly an atlas, on a topological $n$-manifold $M$ is a set $mathcal{A} = {(varphi_j,V_j)mid jin J}$ of charts of $M$ that satisfy the following conditions.
1) The set ${V_jmid jin J}$ is an open cover of $M$, that is, $cup_{jin J}V_j=M$.
2) For every $i$ and $j$ in $J$ such that $V_icap V_jneqemptyset$ we have that $varphi_icirc varphi_j^{-1} : varphi_j(V_icap V_j)to varphi_i(V_icap V_j)$ is smooth, that is, infinitely differentiable.
Remark: Given a topological $n$-manifold $M$ such that there exists an smooth atlas on it, we can consider the poset (partial ordered set) of the set of atlas on $M$ ordered by inclusion.
Definition: A smooth $n$-manifold is a pair $(M,mathcal{A})$, where $M$ is a topological $n$-manifold and $mathcal{A}$ is a maximal smooth atlas on $M$.
Lemma: Let $M$ be a topological $n$-manifold. For every smooth atlas $mathcal{A}$ on $M$ there exists a unique maximal atlas $mathcal{A}^{text{max}}$ on $M$ containing $mathcal{A}$. Given two smooth atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ we have that $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$ if and only if $mathcal{A}_1cupmathcal{A}_2$ is an atlas.
For a proof see Proposition 1.17 in Lee.
Definition: A differential structure on a topological $n$-manifold $M$ is a maximal atlas $mathcal{A}$ on $M$. We say that two atlas $mathcal{A}_1$ and $mathcal{A}_2$ on $M$ determine the same differential structure if $mathcal{A}_1^{text{max}}=mathcal{A}^{text{max}}_2$.
Comments:
1) If you like to adopt an algebraic point of view, we could compare the properties of the poset of atlas on a topological manifold with the poset of topologies on a set and they are very different.
2) The atlas on manifolds say what functions between manifolds are smooth in the same sense that topologies say what functions between topological spaces are continuous. Smooth manifolds and smooth functions form a category and the same is true for topological spaces and continuous functions. In a category we have a notion of isomorphism which says that two objects are equivalent and, as @LeeMosher pointed out in the comments, in many cases it admits an interpretation in terms of preserving the structure. In the first case, an isomorphism is a diffeomorphism and, in the second case, it is an homeomorphism.
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
edited Feb 8 at 1:35
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
answered Feb 2 at 15:48
Dante GrevinoDante Grevino
1,1841112
1,1841112
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
1
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
2
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
|
show 1 more comment
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
1
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
2
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Definition: A differential structure on a topological n-manifold M is $a$ maximal atlas A on M.... I think you missed an $a$ there, or adding it increases readability atleast imo.
$endgroup$
– Maxed
Feb 5 at 7:34
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
$begingroup$
Also in comment 2 I think we want maximal atlases when we consider differentiability? or is the concept relevant for non maximal once?
$endgroup$
– Maxed
Feb 5 at 8:15
1
1
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yes, I missed an "a". Fixed! The point with the fact that every atlas determines a unique differential structure is it allow us to define and talk about a differential structure without worry about the maximal one. We are not able to exhibit a maximal atlas, it is just a theorical concept, but we are able to exhibit an atlas, for example a finite one, in concrete examples.
$endgroup$
– Dante Grevino
Feb 5 at 14:48
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
$begingroup$
Yea, not the finest moment of Zorns Lemma,right? or do we use somthing else when we conclude existance?
$endgroup$
– Maxed
Feb 5 at 14:52
2
2
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
$begingroup$
I think we do not need Zorn Lemma in the proof. Given an atlas, its maximal atlas is simply the set of all compatible charts with the charts of the original atlas. Remain to prove that every two charts there are compatible with each other. It is a nice excercise or you can see Lee's book for details, which is a really great introductory book.
$endgroup$
– Dante Grevino
Feb 5 at 15:00
|
show 1 more comment
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2
$begingroup$
A differential structure is the same as an atlas, or more precisely a maximum atlas. It says what functions defined on the manifold are smooth in the same sense that a topology says what functions are continuous.
$endgroup$
– Dante Grevino
Jan 31 at 5:24
$begingroup$
@DanteGrevino hence it is wrong of me to consider it a more abstract notion? such an group? where two groups can be "algebraically equal" but still very different. A counterexample would be two different familes of charts(atlases) inducing the same differential structure. As long as the bases of the tanget spaces are kind of "similar" I suppose we would get similar differential structure, but that might not be the case.
$endgroup$
– Maxed
Jan 31 at 5:25
2
$begingroup$
Your description of the concept of a "structure" seems to have been imported from algebra. That's not the only concept of "structure" in mathematics. Indeed, mathematicians are willing to accept very different kinds of structure in different branches of mathematics.
$endgroup$
– Lee Mosher
Jan 31 at 15:10
3
$begingroup$
Just like two groups can be isomorphic without being equal, two differentialble manifolds can be diffeomorphic without being equal. The common ground here is the existence of a structure preserving map: a map of groups $f : G to K$ is an isomorphism if it is a bijection and it preserves the group structure (the binary operation and the inversion); a map $f : M to N$ of differentiable manifold is a difffeomorphism if it is a bijection and it preserves the differentiable structure (the pullback of an atlas of $N$ is an atlas of $M$).
$endgroup$
– Lee Mosher
Jan 31 at 15:12
$begingroup$
@LeeMosher I was afraid that might be the case.. stated here ; en.wikipedia.org/wiki/Mathematical_structure
$endgroup$
– Maxed
Jan 31 at 15:13