Source For Partitions of Unity Problems
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I’ve realized that I need to work more with partitions of unity, but unfortunately there are not a lot of problems in the partitions of unity section in Lee which enable you to practice using them, and also get used to some of the tricks one might need to know in their applications. Are there any good sources for partitions of unity problems?
differential-geometry manifolds differential-topology smooth-manifolds
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
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add a comment |
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I’ve realized that I need to work more with partitions of unity, but unfortunately there are not a lot of problems in the partitions of unity section in Lee which enable you to practice using them, and also get used to some of the tricks one might need to know in their applications. Are there any good sources for partitions of unity problems?
differential-geometry manifolds differential-topology smooth-manifolds
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
$endgroup$
$begingroup$
The "Manifolds with Boundary" section in Guilliman and Pollack has an exercise where you construct a function that is >0 in the interior of the manifold, and 0 on the boundary. It is a neat exercise. They are problems 8,10, and 11. Chapter 2 section 1.
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– Prototank
Oct 10 '18 at 1:21
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You can read about Stoke's theorem on manifold, it uses partitions of unity in the proof.
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– Desunkid
Oct 10 '18 at 3:56
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Why don't you try to show that every smooth manifold admits a Riemannian metric, that is, a smooth $(0,2)$-tensor field $g$ on $M$ such that $g_p$ is an inner product on $T_pM$.
$endgroup$
– Matt
Oct 12 '18 at 8:27
add a comment |
$begingroup$
I’ve realized that I need to work more with partitions of unity, but unfortunately there are not a lot of problems in the partitions of unity section in Lee which enable you to practice using them, and also get used to some of the tricks one might need to know in their applications. Are there any good sources for partitions of unity problems?
differential-geometry manifolds differential-topology smooth-manifolds
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
$endgroup$
I’ve realized that I need to work more with partitions of unity, but unfortunately there are not a lot of problems in the partitions of unity section in Lee which enable you to practice using them, and also get used to some of the tricks one might need to know in their applications. Are there any good sources for partitions of unity problems?
differential-geometry manifolds differential-topology smooth-manifolds
differential-geometry manifolds differential-topology smooth-manifolds
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
asked Oct 10 '18 at 0:11
Brandon Thomas Van OverBrandon Thomas Van Over
3,83721127
3,83721127
$begingroup$
The "Manifolds with Boundary" section in Guilliman and Pollack has an exercise where you construct a function that is >0 in the interior of the manifold, and 0 on the boundary. It is a neat exercise. They are problems 8,10, and 11. Chapter 2 section 1.
$endgroup$
– Prototank
Oct 10 '18 at 1:21
$begingroup$
You can read about Stoke's theorem on manifold, it uses partitions of unity in the proof.
$endgroup$
– Desunkid
Oct 10 '18 at 3:56
$begingroup$
Why don't you try to show that every smooth manifold admits a Riemannian metric, that is, a smooth $(0,2)$-tensor field $g$ on $M$ such that $g_p$ is an inner product on $T_pM$.
$endgroup$
– Matt
Oct 12 '18 at 8:27
add a comment |
$begingroup$
The "Manifolds with Boundary" section in Guilliman and Pollack has an exercise where you construct a function that is >0 in the interior of the manifold, and 0 on the boundary. It is a neat exercise. They are problems 8,10, and 11. Chapter 2 section 1.
$endgroup$
– Prototank
Oct 10 '18 at 1:21
$begingroup$
You can read about Stoke's theorem on manifold, it uses partitions of unity in the proof.
$endgroup$
– Desunkid
Oct 10 '18 at 3:56
$begingroup$
Why don't you try to show that every smooth manifold admits a Riemannian metric, that is, a smooth $(0,2)$-tensor field $g$ on $M$ such that $g_p$ is an inner product on $T_pM$.
$endgroup$
– Matt
Oct 12 '18 at 8:27
$begingroup$
The "Manifolds with Boundary" section in Guilliman and Pollack has an exercise where you construct a function that is >0 in the interior of the manifold, and 0 on the boundary. It is a neat exercise. They are problems 8,10, and 11. Chapter 2 section 1.
$endgroup$
– Prototank
Oct 10 '18 at 1:21
$begingroup$
The "Manifolds with Boundary" section in Guilliman and Pollack has an exercise where you construct a function that is >0 in the interior of the manifold, and 0 on the boundary. It is a neat exercise. They are problems 8,10, and 11. Chapter 2 section 1.
$endgroup$
– Prototank
Oct 10 '18 at 1:21
$begingroup$
You can read about Stoke's theorem on manifold, it uses partitions of unity in the proof.
$endgroup$
– Desunkid
Oct 10 '18 at 3:56
$begingroup$
You can read about Stoke's theorem on manifold, it uses partitions of unity in the proof.
$endgroup$
– Desunkid
Oct 10 '18 at 3:56
$begingroup$
Why don't you try to show that every smooth manifold admits a Riemannian metric, that is, a smooth $(0,2)$-tensor field $g$ on $M$ such that $g_p$ is an inner product on $T_pM$.
$endgroup$
– Matt
Oct 12 '18 at 8:27
$begingroup$
Why don't you try to show that every smooth manifold admits a Riemannian metric, that is, a smooth $(0,2)$-tensor field $g$ on $M$ such that $g_p$ is an inner product on $T_pM$.
$endgroup$
– Matt
Oct 12 '18 at 8:27
add a comment |
1 Answer
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One good way to understand partitions of unity and why they are so useful is to study the comparison with the complex case. Take a compact Riemann surface $X$ (maybe $X=mathbb{CP}^1$). A useful object we can define on $X$ is the sheaf of holomorphic functions $mathcal{O}_X$, which assigns to each $Usubseteq X$
$$mathcal{O}_X(U)={f:Uto mathbb{C}:text{holomorphic}}.$$
By the open mapping theorem, a globally defined function $f$ on $X$ is constant because its image would be open and compact in $mathbb{C}$ were it not constant. So, this tells us that global sections $fin mathcal{O}_X(X)$ are just constant functions. Thus, in the compact complex case there are in general very few global objects. In particular, it is hard to take a local object and extend it to a global one in a nontrivial manner.
Over $mathbb{R}$, the situation is vastly different, due essentially to the existence of bump functions. A bump function is a compactly supported $mathcal{C}^infty$ function on $mathbb{R}^n$. Because holomorphicity is very restrictive, such functions do not exist in $mathcal{H}(mathbb{C})$. Bump functions (and by extension partitions of unity) allow us to extend local objects defined in coordinates to global objects, namely global sections. So by contrast, given a compact manifold $M$, $mathcal{C}^infty(M)$ has unimaginably many elements. The fact that we can extend local objects by zero using bump functions lets us construct many global objects that are very important.
As for resources, try Tu's Introduction to Manifolds, Bott and Tu, or maybe Guillemin and Pollack. If you can't find enough exercises on this, why not study the proof of the existence of Riemannian metrics on smooth manifolds, and see what other kinds of objects you can come up with.
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
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add a comment |
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$begingroup$
One good way to understand partitions of unity and why they are so useful is to study the comparison with the complex case. Take a compact Riemann surface $X$ (maybe $X=mathbb{CP}^1$). A useful object we can define on $X$ is the sheaf of holomorphic functions $mathcal{O}_X$, which assigns to each $Usubseteq X$
$$mathcal{O}_X(U)={f:Uto mathbb{C}:text{holomorphic}}.$$
By the open mapping theorem, a globally defined function $f$ on $X$ is constant because its image would be open and compact in $mathbb{C}$ were it not constant. So, this tells us that global sections $fin mathcal{O}_X(X)$ are just constant functions. Thus, in the compact complex case there are in general very few global objects. In particular, it is hard to take a local object and extend it to a global one in a nontrivial manner.
Over $mathbb{R}$, the situation is vastly different, due essentially to the existence of bump functions. A bump function is a compactly supported $mathcal{C}^infty$ function on $mathbb{R}^n$. Because holomorphicity is very restrictive, such functions do not exist in $mathcal{H}(mathbb{C})$. Bump functions (and by extension partitions of unity) allow us to extend local objects defined in coordinates to global objects, namely global sections. So by contrast, given a compact manifold $M$, $mathcal{C}^infty(M)$ has unimaginably many elements. The fact that we can extend local objects by zero using bump functions lets us construct many global objects that are very important.
As for resources, try Tu's Introduction to Manifolds, Bott and Tu, or maybe Guillemin and Pollack. If you can't find enough exercises on this, why not study the proof of the existence of Riemannian metrics on smooth manifolds, and see what other kinds of objects you can come up with.
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
$endgroup$
add a comment |
$begingroup$
One good way to understand partitions of unity and why they are so useful is to study the comparison with the complex case. Take a compact Riemann surface $X$ (maybe $X=mathbb{CP}^1$). A useful object we can define on $X$ is the sheaf of holomorphic functions $mathcal{O}_X$, which assigns to each $Usubseteq X$
$$mathcal{O}_X(U)={f:Uto mathbb{C}:text{holomorphic}}.$$
By the open mapping theorem, a globally defined function $f$ on $X$ is constant because its image would be open and compact in $mathbb{C}$ were it not constant. So, this tells us that global sections $fin mathcal{O}_X(X)$ are just constant functions. Thus, in the compact complex case there are in general very few global objects. In particular, it is hard to take a local object and extend it to a global one in a nontrivial manner.
Over $mathbb{R}$, the situation is vastly different, due essentially to the existence of bump functions. A bump function is a compactly supported $mathcal{C}^infty$ function on $mathbb{R}^n$. Because holomorphicity is very restrictive, such functions do not exist in $mathcal{H}(mathbb{C})$. Bump functions (and by extension partitions of unity) allow us to extend local objects defined in coordinates to global objects, namely global sections. So by contrast, given a compact manifold $M$, $mathcal{C}^infty(M)$ has unimaginably many elements. The fact that we can extend local objects by zero using bump functions lets us construct many global objects that are very important.
As for resources, try Tu's Introduction to Manifolds, Bott and Tu, or maybe Guillemin and Pollack. If you can't find enough exercises on this, why not study the proof of the existence of Riemannian metrics on smooth manifolds, and see what other kinds of objects you can come up with.
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
$endgroup$
add a comment |
$begingroup$
One good way to understand partitions of unity and why they are so useful is to study the comparison with the complex case. Take a compact Riemann surface $X$ (maybe $X=mathbb{CP}^1$). A useful object we can define on $X$ is the sheaf of holomorphic functions $mathcal{O}_X$, which assigns to each $Usubseteq X$
$$mathcal{O}_X(U)={f:Uto mathbb{C}:text{holomorphic}}.$$
By the open mapping theorem, a globally defined function $f$ on $X$ is constant because its image would be open and compact in $mathbb{C}$ were it not constant. So, this tells us that global sections $fin mathcal{O}_X(X)$ are just constant functions. Thus, in the compact complex case there are in general very few global objects. In particular, it is hard to take a local object and extend it to a global one in a nontrivial manner.
Over $mathbb{R}$, the situation is vastly different, due essentially to the existence of bump functions. A bump function is a compactly supported $mathcal{C}^infty$ function on $mathbb{R}^n$. Because holomorphicity is very restrictive, such functions do not exist in $mathcal{H}(mathbb{C})$. Bump functions (and by extension partitions of unity) allow us to extend local objects defined in coordinates to global objects, namely global sections. So by contrast, given a compact manifold $M$, $mathcal{C}^infty(M)$ has unimaginably many elements. The fact that we can extend local objects by zero using bump functions lets us construct many global objects that are very important.
As for resources, try Tu's Introduction to Manifolds, Bott and Tu, or maybe Guillemin and Pollack. If you can't find enough exercises on this, why not study the proof of the existence of Riemannian metrics on smooth manifolds, and see what other kinds of objects you can come up with.
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
$endgroup$
One good way to understand partitions of unity and why they are so useful is to study the comparison with the complex case. Take a compact Riemann surface $X$ (maybe $X=mathbb{CP}^1$). A useful object we can define on $X$ is the sheaf of holomorphic functions $mathcal{O}_X$, which assigns to each $Usubseteq X$
$$mathcal{O}_X(U)={f:Uto mathbb{C}:text{holomorphic}}.$$
By the open mapping theorem, a globally defined function $f$ on $X$ is constant because its image would be open and compact in $mathbb{C}$ were it not constant. So, this tells us that global sections $fin mathcal{O}_X(X)$ are just constant functions. Thus, in the compact complex case there are in general very few global objects. In particular, it is hard to take a local object and extend it to a global one in a nontrivial manner.
Over $mathbb{R}$, the situation is vastly different, due essentially to the existence of bump functions. A bump function is a compactly supported $mathcal{C}^infty$ function on $mathbb{R}^n$. Because holomorphicity is very restrictive, such functions do not exist in $mathcal{H}(mathbb{C})$. Bump functions (and by extension partitions of unity) allow us to extend local objects defined in coordinates to global objects, namely global sections. So by contrast, given a compact manifold $M$, $mathcal{C}^infty(M)$ has unimaginably many elements. The fact that we can extend local objects by zero using bump functions lets us construct many global objects that are very important.
As for resources, try Tu's Introduction to Manifolds, Bott and Tu, or maybe Guillemin and Pollack. If you can't find enough exercises on this, why not study the proof of the existence of Riemannian metrics on smooth manifolds, and see what other kinds of objects you can come up with.
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
answered Dec 20 '18 at 18:57
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
9,91741640
9,91741640
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$begingroup$
The "Manifolds with Boundary" section in Guilliman and Pollack has an exercise where you construct a function that is >0 in the interior of the manifold, and 0 on the boundary. It is a neat exercise. They are problems 8,10, and 11. Chapter 2 section 1.
$endgroup$
– Prototank
Oct 10 '18 at 1:21
$begingroup$
You can read about Stoke's theorem on manifold, it uses partitions of unity in the proof.
$endgroup$
– Desunkid
Oct 10 '18 at 3:56
$begingroup$
Why don't you try to show that every smooth manifold admits a Riemannian metric, that is, a smooth $(0,2)$-tensor field $g$ on $M$ such that $g_p$ is an inner product on $T_pM$.
$endgroup$
– Matt
Oct 12 '18 at 8:27