Which properties preserve by isometry?
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0
down vote
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I know that an isometry between two surfaces preserves the 1st fundamental coefficients, geodesic curvature, and Gaussian curvature.
I wonder that how about the curvature and torsion of a curve on the surface, and 2nd fundamental coefficients?
I can’t find any counterexample and prove this question
Give some advice or comments! Thank you!
differential-geometry curvature isometry
asked 2 days ago
Primavera
2389
add a comment |
up vote
0
down vote
favorite
I know that an isometry between two surfaces preserves the 1st fundamental coefficients, geodesic curvature, and Gaussian curvature.
I wonder that how about the curvature and torsion of a curve on the surface, and 2nd fundamental coefficients?
I can’t find any counterexample and prove this question
Give some advice or comments! Thank you!
differential-geometry curvature isometry
asked 2 days ago
Primavera
2389
Once the metric is preserved, every thing depends completely on it is.
– Semsem
2 days ago
1
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image?
– Travis
2 days ago
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
yesterday
1
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I know that an isometry between two surfaces preserves the 1st fundamental coefficients, geodesic curvature, and Gaussian curvature.
I wonder that how about the curvature and torsion of a curve on the surface, and 2nd fundamental coefficients?
I can’t find any counterexample and prove this question
Give some advice or comments! Thank you!
differential-geometry curvature isometry
asked 2 days ago
Primavera
2389
I know that an isometry between two surfaces preserves the 1st fundamental coefficients, geodesic curvature, and Gaussian curvature.
I wonder that how about the curvature and torsion of a curve on the surface, and 2nd fundamental coefficients?
I can’t find any counterexample and prove this question
Give some advice or comments! Thank you!
differential-geometry curvature isometry
differential-geometry curvature isometry
asked 2 days ago
Primavera
2389
asked 2 days ago
Primavera
2389
asked 2 days ago
Primavera
2389
asked 2 days ago
Primavera
2389
asked 2 days ago
Primavera
2389
2389
Once the metric is preserved, every thing depends completely on it is.
– Semsem
2 days ago
1
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image?
– Travis
2 days ago
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
yesterday
1
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
yesterday
add a comment |
Once the metric is preserved, every thing depends completely on it is.
– Semsem
2 days ago
1
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image?
– Travis
2 days ago
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
yesterday
1
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
yesterday
Once the metric is preserved, every thing depends completely on it is.
– Semsem
2 days ago
Once the metric is preserved, every thing depends completely on it is.
– Semsem
2 days ago
1
1
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image?
– Travis
2 days ago
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image?
– Travis
2 days ago
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
yesterday
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
yesterday
1
1
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
yesterday
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
yesterday
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
Hint Consider an isometry from a subset of a plane in $Bbb R^3$ to a subset of a cylinder in $Bbb R^3$.
answered yesterday
Travis
58.7k765142
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
|
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint Consider an isometry from a subset of a plane in $Bbb R^3$ to a subset of a cylinder in $Bbb R^3$.
answered yesterday
Travis
58.7k765142
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
|
show 1 more comment
up vote
1
down vote
Hint Consider an isometry from a subset of a plane in $Bbb R^3$ to a subset of a cylinder in $Bbb R^3$.
answered yesterday
Travis
58.7k765142
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
|
show 1 more comment
up vote
1
down vote
up vote
1
down vote
Hint Consider an isometry from a subset of a plane in $Bbb R^3$ to a subset of a cylinder in $Bbb R^3$.
answered yesterday
Travis
58.7k765142
Hint Consider an isometry from a subset of a plane in $Bbb R^3$ to a subset of a cylinder in $Bbb R^3$.
answered yesterday
Travis
58.7k765142
answered yesterday
Travis
58.7k765142
answered yesterday
Travis
58.7k765142
answered yesterday
Travis
58.7k765142
58.7k765142
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
|
show 1 more comment
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right?
– Primavera
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.)
– Travis
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients.
– Primavera
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
What are the second fundamental forms for the two surfaces in the hint?
– Travis
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
22 hours ago
|
show 1 more comment
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Once the metric is preserved, every thing depends completely on it is.
– Semsem
2 days ago
1
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image?
– Travis
2 days ago
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
yesterday
1
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
yesterday