Which properties preserve by isometry?
up vote
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 Nov 18 at 10:01
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 Nov 18 at 10:01
Primavera
2389
Once the metric is preserved, every thing depends completely on it is.
– Semsem
Nov 18 at 15:09
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
Nov 18 at 16:04
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
Nov 19 at 6:01
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
Nov 19 at 14:40
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 Nov 18 at 10:01
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 Nov 18 at 10:01
Primavera
2389
asked Nov 18 at 10:01
Primavera
2389
asked Nov 18 at 10:01
Primavera
2389
asked Nov 18 at 10:01
Primavera
2389
asked Nov 18 at 10:01
Primavera
2389
2389
Once the metric is preserved, every thing depends completely on it is.
– Semsem
Nov 18 at 15:09
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
Nov 18 at 16:04
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
Nov 19 at 6:01
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
Nov 19 at 14:40
add a comment |
Once the metric is preserved, every thing depends completely on it is.
– Semsem
Nov 18 at 15:09
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
Nov 18 at 16:04
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
Nov 19 at 6:01
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
Nov 19 at 14:40
Once the metric is preserved, every thing depends completely on it is.
– Semsem
Nov 18 at 15:09
Once the metric is preserved, every thing depends completely on it is.
– Semsem
Nov 18 at 15:09
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
Nov 18 at 16:04
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
Nov 18 at 16:04
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
Nov 19 at 6:01
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
Nov 19 at 6:01
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
Nov 19 at 14:40
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $Bbb R^3$.
– Travis
Nov 19 at 14:40
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 Nov 19 at 14:40
Travis
59k766144
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
Nov 20 at 6:20
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
Nov 20 at 6:24
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
Nov 20 at 6:40
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
|
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 Nov 19 at 14:40
Travis
59k766144
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
Nov 20 at 6:20
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
Nov 20 at 6:24
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
Nov 20 at 6:40
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
|
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 Nov 19 at 14:40
Travis
59k766144
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
Nov 20 at 6:20
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
Nov 20 at 6:24
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
Nov 20 at 6:40
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
|
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 Nov 19 at 14:40
Travis
59k766144
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 Nov 19 at 14:40
Travis
59k766144
answered Nov 19 at 14:40
Travis
59k766144
answered Nov 19 at 14:40
Travis
59k766144
answered Nov 19 at 14:40
Travis
59k766144
59k766144
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
Nov 20 at 6:20
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
Nov 20 at 6:24
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
Nov 20 at 6:40
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
|
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
Nov 20 at 6:20
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
Nov 20 at 6:24
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
Nov 20 at 6:40
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
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
Nov 20 at 6:20
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
Nov 20 at 6:20
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
Nov 20 at 6:24
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
Nov 20 at 6:24
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
Nov 20 at 6:40
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
Nov 20 at 6:40
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
What are the second fundamental forms for the two surfaces in the hint?
– Travis
Nov 20 at 6:41
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
Nov 20 at 6:50
|
show 1 more comment
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Once the metric is preserved, every thing depends completely on it is.
– Semsem
Nov 18 at 15:09
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
Nov 18 at 16:04
@Travis Just elementary differential geometry level, it means 'in $mathbb{R}^{3}$ Euclidean space' case.
– Primavera
Nov 19 at 6:01
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
Nov 19 at 14:40