The need for a certain property in a proof of a basic fact about Lie derivatives
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Below is a proof from Lee's Smooth Manifolds, 2nd ed.
I don't see where the fact that the set of regular points is open is used in the proof. It seems never to be invoked.
I'd appreciate it if someone could either confirm this or point to where that fact is used.
UPDATE: I don't believe that the answer is the first sentence of Case 1, as this is what Theorem 9.22 says.
general-topology differential-geometry proof-explanation differential-topology
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
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add a comment |
$begingroup$
Below is a proof from Lee's Smooth Manifolds, 2nd ed.
I don't see where the fact that the set of regular points is open is used in the proof. It seems never to be invoked.
I'd appreciate it if someone could either confirm this or point to where that fact is used.
UPDATE: I don't believe that the answer is the first sentence of Case 1, as this is what Theorem 9.22 says.
general-topology differential-geometry proof-explanation differential-topology
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
$endgroup$
1
$begingroup$
The proof works fine without the statement that $mathscr R(V)$ is open. I think I included that remark just as an aid to visualizing what's going on.
$endgroup$
– Jack Lee
Dec 16 '18 at 1:46
add a comment |
$begingroup$
Below is a proof from Lee's Smooth Manifolds, 2nd ed.
I don't see where the fact that the set of regular points is open is used in the proof. It seems never to be invoked.
I'd appreciate it if someone could either confirm this or point to where that fact is used.
UPDATE: I don't believe that the answer is the first sentence of Case 1, as this is what Theorem 9.22 says.
general-topology differential-geometry proof-explanation differential-topology
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
$endgroup$
Below is a proof from Lee's Smooth Manifolds, 2nd ed.
I don't see where the fact that the set of regular points is open is used in the proof. It seems never to be invoked.
I'd appreciate it if someone could either confirm this or point to where that fact is used.
UPDATE: I don't believe that the answer is the first sentence of Case 1, as this is what Theorem 9.22 says.
general-topology differential-geometry proof-explanation differential-topology
general-topology differential-geometry proof-explanation differential-topology
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
edited Dec 14 '18 at 20:17
CuriousKid7
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
asked Dec 14 '18 at 9:34
CuriousKid7CuriousKid7
1,664717
1,664717
1
$begingroup$
The proof works fine without the statement that $mathscr R(V)$ is open. I think I included that remark just as an aid to visualizing what's going on.
$endgroup$
– Jack Lee
Dec 16 '18 at 1:46
add a comment |
1
$begingroup$
The proof works fine without the statement that $mathscr R(V)$ is open. I think I included that remark just as an aid to visualizing what's going on.
$endgroup$
– Jack Lee
Dec 16 '18 at 1:46
1
1
$begingroup$
The proof works fine without the statement that $mathscr R(V)$ is open. I think I included that remark just as an aid to visualizing what's going on.
$endgroup$
– Jack Lee
Dec 16 '18 at 1:46
$begingroup$
The proof works fine without the statement that $mathscr R(V)$ is open. I think I included that remark just as an aid to visualizing what's going on.
$endgroup$
– Jack Lee
Dec 16 '18 at 1:46
add a comment |
1 Answer
1
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In the first sentence of case 1: a neighbourhood of $p$ such that... This neighbourhood must be a subset of $mathcal{R} (V)$.
answered Dec 14 '18 at 10:14
user626368user626368
191
$endgroup$
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
add a comment |
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1 Answer
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$begingroup$
In the first sentence of case 1: a neighbourhood of $p$ such that... This neighbourhood must be a subset of $mathcal{R} (V)$.
answered Dec 14 '18 at 10:14
user626368user626368
191
$endgroup$
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
add a comment |
$begingroup$
In the first sentence of case 1: a neighbourhood of $p$ such that... This neighbourhood must be a subset of $mathcal{R} (V)$.
answered Dec 14 '18 at 10:14
user626368user626368
191
$endgroup$
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
add a comment |
$begingroup$
In the first sentence of case 1: a neighbourhood of $p$ such that... This neighbourhood must be a subset of $mathcal{R} (V)$.
answered Dec 14 '18 at 10:14
user626368user626368
191
$endgroup$
In the first sentence of case 1: a neighbourhood of $p$ such that... This neighbourhood must be a subset of $mathcal{R} (V)$.
answered Dec 14 '18 at 10:14
user626368user626368
191
answered Dec 14 '18 at 10:14
user626368user626368
191
answered Dec 14 '18 at 10:14
user626368user626368
191
answered Dec 14 '18 at 10:14
user626368user626368
191
191
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
add a comment |
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
$begingroup$
I don't think so. See my update.
$endgroup$
– CuriousKid7
Dec 14 '18 at 18:38
add a comment |
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1
$begingroup$
The proof works fine without the statement that $mathscr R(V)$ is open. I think I included that remark just as an aid to visualizing what's going on.
$endgroup$
– Jack Lee
Dec 16 '18 at 1:46