Which properties preserve by isometry?











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







share|cite|improve this question






















  • 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















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







share|cite|improve this question






















  • 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













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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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


















  • 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










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 $Bbb R^3$.






share|cite|improve this answer





















  • 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
    19 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
    19 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
    18 hours ago












  • What are the second fundamental forms for the two surfaces in the hint?
    – Travis
    18 hours ago










  • Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
    – Primavera
    18 hours ago











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














 

draft saved


draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003337%2fwhich-properties-preserve-by-isometry%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























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 $Bbb R^3$.






share|cite|improve this answer





















  • 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
    19 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
    19 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
    18 hours ago












  • What are the second fundamental forms for the two surfaces in the hint?
    – Travis
    18 hours ago










  • Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
    – Primavera
    18 hours ago















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 $Bbb R^3$.






share|cite|improve this answer





















  • 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
    19 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
    19 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
    18 hours ago












  • What are the second fundamental forms for the two surfaces in the hint?
    – Travis
    18 hours ago










  • Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
    – Primavera
    18 hours ago













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 $Bbb R^3$.






share|cite|improve this answer












Hint Consider an isometry from a subset of a plane in $Bbb R^3$ to a subset of a cylinder in $Bbb R^3$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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
    19 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
    19 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
    18 hours ago












  • What are the second fundamental forms for the two surfaces in the hint?
    – Travis
    18 hours ago










  • Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
    – Primavera
    18 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
    19 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
    19 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
    18 hours ago












  • What are the second fundamental forms for the two surfaces in the hint?
    – Travis
    18 hours ago










  • Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
    – Primavera
    18 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
19 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
19 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
19 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
19 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
18 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
18 hours ago














What are the second fundamental forms for the two surfaces in the hint?
– Travis
18 hours ago




What are the second fundamental forms for the two surfaces in the hint?
– Travis
18 hours ago












Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
18 hours ago




Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear!
– Primavera
18 hours ago


















 

draft saved


draft discarded



















































 


draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003337%2fwhich-properties-preserve-by-isometry%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

Image
28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
Read more

Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

Image
up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
Read more

A Topological Invariant for $pi_3(U(n))$

Image
3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
Read more