How to transfer the integration theory of smooth manifold to that of $C^k$ manifold
$begingroup$
I want to consider $C^k$ manifolds because, in PDE, people want to consider $C^k$ boundaries etc, and a version of divergence theorem for $C^k$ manifold is needed. Is it necessary to repeat all the proofs for $C^k$ manifolds? How to think of it in terms of the theory of smooth manifolds?
I think there can be a lot of fundamental differences between the two. For starters, the tangent space of a $C^k$ manifold is not of finite dimension if we use derivation to define it. Also, I am not sure whether there is a $C^k$ partition of unity for $C^k$ a manifold. So that makes me wonder whether it is a good idea to think of $C^k$ manifolds in terms of smooth ones, then just skip
all the proofs.
Can someone familiar with PDE give me some hints on how people think of divergence theorems for $C^k$ boundaries? I know that there are ways that you can derive the divergence theorem without defining tangent spaces etc. but ignoring the similarity between it and the theory of smooth manifolds makes me feel so uncomfortable.
differential-geometry pde differential-topology smooth-manifolds
asked Jan 3 at 5:16
KeithKeith
420317
$endgroup$
|
show 3 more comments
$begingroup$
I want to consider $C^k$ manifolds because, in PDE, people want to consider $C^k$ boundaries etc, and a version of divergence theorem for $C^k$ manifold is needed. Is it necessary to repeat all the proofs for $C^k$ manifolds? How to think of it in terms of the theory of smooth manifolds?
I think there can be a lot of fundamental differences between the two. For starters, the tangent space of a $C^k$ manifold is not of finite dimension if we use derivation to define it. Also, I am not sure whether there is a $C^k$ partition of unity for $C^k$ a manifold. So that makes me wonder whether it is a good idea to think of $C^k$ manifolds in terms of smooth ones, then just skip
all the proofs.
Can someone familiar with PDE give me some hints on how people think of divergence theorems for $C^k$ boundaries? I know that there are ways that you can derive the divergence theorem without defining tangent spaces etc. but ignoring the similarity between it and the theory of smooth manifolds makes me feel so uncomfortable.
differential-geometry pde differential-topology smooth-manifolds
asked Jan 3 at 5:16
KeithKeith
420317
$endgroup$
$begingroup$
Every $C^k$ manifold, with $kgeq 1$, has a smooth atlas. See e.g. here. But it can be useful to consider non-smooth objects on a smooth manifold, eg functions, vector fields, differential forms, cycles, etc. Many theorems do not require smooth objects (e.g., Stokes Theorem). You can check the proof of each result you need to see what is the degree of regularity needed.
$endgroup$
– Gil Bor
Jan 3 at 5:33
$begingroup$
The proofs are almost always essentially identical. The point you observe about tangent spaces doesn't change much, it just says "derivations on functions" is not the right definition anymore (and you won't see it outside a book on smooth manifolds specifically). There are some cases where the result actually changes (Sard's theorem) but largely nothing really does.
$endgroup$
– Mike Miller
Jan 3 at 13:58
$begingroup$
Do you mean that in the C^k case we only need to adapt the definition so that the tangent space is finite dimensional? May I ask whether there is any reference on this matter?
$endgroup$
– Keith
Jan 3 at 15:51
$begingroup$
How do you use that result to transfer the translate the results of smooth manifolds to $C^k$ manifolds? How would you define $df$ for a $C^k$ function? @GilBor
$endgroup$
– Keith
Jan 4 at 5:37
$begingroup$
@Keith Yes, for $C^k$ manifolds you have two choices, both of which are also valid in the $C^infty$ case. 1) Define the tangent bundle by picking an atlas and over each chart, replacing $U subset Bbb R^n$ with $U times Bbb R^n$; the vector bundle transition functions (what happens when you change charts) act on the $Bbb R^n$ factor by the Jacobian of the manifold-level transition maps between charts. 2) Define $T_x M$ to be the set of $gamma'(0)$ for $gamma: (-t, t) to M$ a $C^k$ curve with $gamma(0) = 0$. The vector space operations are defined as in (1).
$endgroup$
– Mike Miller
Jan 4 at 16:14
|
show 3 more comments
$begingroup$
I want to consider $C^k$ manifolds because, in PDE, people want to consider $C^k$ boundaries etc, and a version of divergence theorem for $C^k$ manifold is needed. Is it necessary to repeat all the proofs for $C^k$ manifolds? How to think of it in terms of the theory of smooth manifolds?
I think there can be a lot of fundamental differences between the two. For starters, the tangent space of a $C^k$ manifold is not of finite dimension if we use derivation to define it. Also, I am not sure whether there is a $C^k$ partition of unity for $C^k$ a manifold. So that makes me wonder whether it is a good idea to think of $C^k$ manifolds in terms of smooth ones, then just skip
all the proofs.
Can someone familiar with PDE give me some hints on how people think of divergence theorems for $C^k$ boundaries? I know that there are ways that you can derive the divergence theorem without defining tangent spaces etc. but ignoring the similarity between it and the theory of smooth manifolds makes me feel so uncomfortable.
differential-geometry pde differential-topology smooth-manifolds
asked Jan 3 at 5:16
KeithKeith
420317
$endgroup$
I want to consider $C^k$ manifolds because, in PDE, people want to consider $C^k$ boundaries etc, and a version of divergence theorem for $C^k$ manifold is needed. Is it necessary to repeat all the proofs for $C^k$ manifolds? How to think of it in terms of the theory of smooth manifolds?
I think there can be a lot of fundamental differences between the two. For starters, the tangent space of a $C^k$ manifold is not of finite dimension if we use derivation to define it. Also, I am not sure whether there is a $C^k$ partition of unity for $C^k$ a manifold. So that makes me wonder whether it is a good idea to think of $C^k$ manifolds in terms of smooth ones, then just skip
all the proofs.
Can someone familiar with PDE give me some hints on how people think of divergence theorems for $C^k$ boundaries? I know that there are ways that you can derive the divergence theorem without defining tangent spaces etc. but ignoring the similarity between it and the theory of smooth manifolds makes me feel so uncomfortable.
differential-geometry pde differential-topology smooth-manifolds
differential-geometry pde differential-topology smooth-manifolds
asked Jan 3 at 5:16
KeithKeith
420317
asked Jan 3 at 5:16
KeithKeith
420317
edited Jan 4 at 5:41
Keith
asked Jan 3 at 5:16
KeithKeith
420317
asked Jan 3 at 5:16
KeithKeith
420317
asked Jan 3 at 5:16
KeithKeith
420317
420317
$begingroup$
Every $C^k$ manifold, with $kgeq 1$, has a smooth atlas. See e.g. here. But it can be useful to consider non-smooth objects on a smooth manifold, eg functions, vector fields, differential forms, cycles, etc. Many theorems do not require smooth objects (e.g., Stokes Theorem). You can check the proof of each result you need to see what is the degree of regularity needed.
$endgroup$
– Gil Bor
Jan 3 at 5:33
$begingroup$
The proofs are almost always essentially identical. The point you observe about tangent spaces doesn't change much, it just says "derivations on functions" is not the right definition anymore (and you won't see it outside a book on smooth manifolds specifically). There are some cases where the result actually changes (Sard's theorem) but largely nothing really does.
$endgroup$
– Mike Miller
Jan 3 at 13:58
$begingroup$
Do you mean that in the C^k case we only need to adapt the definition so that the tangent space is finite dimensional? May I ask whether there is any reference on this matter?
$endgroup$
– Keith
Jan 3 at 15:51
$begingroup$
How do you use that result to transfer the translate the results of smooth manifolds to $C^k$ manifolds? How would you define $df$ for a $C^k$ function? @GilBor
$endgroup$
– Keith
Jan 4 at 5:37
$begingroup$
@Keith Yes, for $C^k$ manifolds you have two choices, both of which are also valid in the $C^infty$ case. 1) Define the tangent bundle by picking an atlas and over each chart, replacing $U subset Bbb R^n$ with $U times Bbb R^n$; the vector bundle transition functions (what happens when you change charts) act on the $Bbb R^n$ factor by the Jacobian of the manifold-level transition maps between charts. 2) Define $T_x M$ to be the set of $gamma'(0)$ for $gamma: (-t, t) to M$ a $C^k$ curve with $gamma(0) = 0$. The vector space operations are defined as in (1).
$endgroup$
– Mike Miller
Jan 4 at 16:14
|
show 3 more comments
$begingroup$
Every $C^k$ manifold, with $kgeq 1$, has a smooth atlas. See e.g. here. But it can be useful to consider non-smooth objects on a smooth manifold, eg functions, vector fields, differential forms, cycles, etc. Many theorems do not require smooth objects (e.g., Stokes Theorem). You can check the proof of each result you need to see what is the degree of regularity needed.
$endgroup$
– Gil Bor
Jan 3 at 5:33
$begingroup$
The proofs are almost always essentially identical. The point you observe about tangent spaces doesn't change much, it just says "derivations on functions" is not the right definition anymore (and you won't see it outside a book on smooth manifolds specifically). There are some cases where the result actually changes (Sard's theorem) but largely nothing really does.
$endgroup$
– Mike Miller
Jan 3 at 13:58
$begingroup$
Do you mean that in the C^k case we only need to adapt the definition so that the tangent space is finite dimensional? May I ask whether there is any reference on this matter?
$endgroup$
– Keith
Jan 3 at 15:51
$begingroup$
How do you use that result to transfer the translate the results of smooth manifolds to $C^k$ manifolds? How would you define $df$ for a $C^k$ function? @GilBor
$endgroup$
– Keith
Jan 4 at 5:37
$begingroup$
@Keith Yes, for $C^k$ manifolds you have two choices, both of which are also valid in the $C^infty$ case. 1) Define the tangent bundle by picking an atlas and over each chart, replacing $U subset Bbb R^n$ with $U times Bbb R^n$; the vector bundle transition functions (what happens when you change charts) act on the $Bbb R^n$ factor by the Jacobian of the manifold-level transition maps between charts. 2) Define $T_x M$ to be the set of $gamma'(0)$ for $gamma: (-t, t) to M$ a $C^k$ curve with $gamma(0) = 0$. The vector space operations are defined as in (1).
$endgroup$
– Mike Miller
Jan 4 at 16:14
$begingroup$
Every $C^k$ manifold, with $kgeq 1$, has a smooth atlas. See e.g. here. But it can be useful to consider non-smooth objects on a smooth manifold, eg functions, vector fields, differential forms, cycles, etc. Many theorems do not require smooth objects (e.g., Stokes Theorem). You can check the proof of each result you need to see what is the degree of regularity needed.
$endgroup$
– Gil Bor
Jan 3 at 5:33
$begingroup$
Every $C^k$ manifold, with $kgeq 1$, has a smooth atlas. See e.g. here. But it can be useful to consider non-smooth objects on a smooth manifold, eg functions, vector fields, differential forms, cycles, etc. Many theorems do not require smooth objects (e.g., Stokes Theorem). You can check the proof of each result you need to see what is the degree of regularity needed.
$endgroup$
– Gil Bor
Jan 3 at 5:33
$begingroup$
The proofs are almost always essentially identical. The point you observe about tangent spaces doesn't change much, it just says "derivations on functions" is not the right definition anymore (and you won't see it outside a book on smooth manifolds specifically). There are some cases where the result actually changes (Sard's theorem) but largely nothing really does.
$endgroup$
– Mike Miller
Jan 3 at 13:58
$begingroup$
The proofs are almost always essentially identical. The point you observe about tangent spaces doesn't change much, it just says "derivations on functions" is not the right definition anymore (and you won't see it outside a book on smooth manifolds specifically). There are some cases where the result actually changes (Sard's theorem) but largely nothing really does.
$endgroup$
– Mike Miller
Jan 3 at 13:58
$begingroup$
Do you mean that in the C^k case we only need to adapt the definition so that the tangent space is finite dimensional? May I ask whether there is any reference on this matter?
$endgroup$
– Keith
Jan 3 at 15:51
$begingroup$
Do you mean that in the C^k case we only need to adapt the definition so that the tangent space is finite dimensional? May I ask whether there is any reference on this matter?
$endgroup$
– Keith
Jan 3 at 15:51
$begingroup$
How do you use that result to transfer the translate the results of smooth manifolds to $C^k$ manifolds? How would you define $df$ for a $C^k$ function? @GilBor
$endgroup$
– Keith
Jan 4 at 5:37
$begingroup$
How do you use that result to transfer the translate the results of smooth manifolds to $C^k$ manifolds? How would you define $df$ for a $C^k$ function? @GilBor
$endgroup$
– Keith
Jan 4 at 5:37
$begingroup$
@Keith Yes, for $C^k$ manifolds you have two choices, both of which are also valid in the $C^infty$ case. 1) Define the tangent bundle by picking an atlas and over each chart, replacing $U subset Bbb R^n$ with $U times Bbb R^n$; the vector bundle transition functions (what happens when you change charts) act on the $Bbb R^n$ factor by the Jacobian of the manifold-level transition maps between charts. 2) Define $T_x M$ to be the set of $gamma'(0)$ for $gamma: (-t, t) to M$ a $C^k$ curve with $gamma(0) = 0$. The vector space operations are defined as in (1).
$endgroup$
– Mike Miller
Jan 4 at 16:14
$begingroup$
@Keith Yes, for $C^k$ manifolds you have two choices, both of which are also valid in the $C^infty$ case. 1) Define the tangent bundle by picking an atlas and over each chart, replacing $U subset Bbb R^n$ with $U times Bbb R^n$; the vector bundle transition functions (what happens when you change charts) act on the $Bbb R^n$ factor by the Jacobian of the manifold-level transition maps between charts. 2) Define $T_x M$ to be the set of $gamma'(0)$ for $gamma: (-t, t) to M$ a $C^k$ curve with $gamma(0) = 0$. The vector space operations are defined as in (1).
$endgroup$
– Mike Miller
Jan 4 at 16:14
|
show 3 more comments
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$begingroup$
Every $C^k$ manifold, with $kgeq 1$, has a smooth atlas. See e.g. here. But it can be useful to consider non-smooth objects on a smooth manifold, eg functions, vector fields, differential forms, cycles, etc. Many theorems do not require smooth objects (e.g., Stokes Theorem). You can check the proof of each result you need to see what is the degree of regularity needed.
$endgroup$
– Gil Bor
Jan 3 at 5:33
$begingroup$
The proofs are almost always essentially identical. The point you observe about tangent spaces doesn't change much, it just says "derivations on functions" is not the right definition anymore (and you won't see it outside a book on smooth manifolds specifically). There are some cases where the result actually changes (Sard's theorem) but largely nothing really does.
$endgroup$
– Mike Miller
Jan 3 at 13:58
$begingroup$
Do you mean that in the C^k case we only need to adapt the definition so that the tangent space is finite dimensional? May I ask whether there is any reference on this matter?
$endgroup$
– Keith
Jan 3 at 15:51
$begingroup$
How do you use that result to transfer the translate the results of smooth manifolds to $C^k$ manifolds? How would you define $df$ for a $C^k$ function? @GilBor
$endgroup$
– Keith
Jan 4 at 5:37
$begingroup$
@Keith Yes, for $C^k$ manifolds you have two choices, both of which are also valid in the $C^infty$ case. 1) Define the tangent bundle by picking an atlas and over each chart, replacing $U subset Bbb R^n$ with $U times Bbb R^n$; the vector bundle transition functions (what happens when you change charts) act on the $Bbb R^n$ factor by the Jacobian of the manifold-level transition maps between charts. 2) Define $T_x M$ to be the set of $gamma'(0)$ for $gamma: (-t, t) to M$ a $C^k$ curve with $gamma(0) = 0$. The vector space operations are defined as in (1).
$endgroup$
– Mike Miller
Jan 4 at 16:14