Isometric embedding of a genus g surface












5












$begingroup$


Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $mathbb{R}^4?$






dg.differential-geometry riemannian-geometry riemann-surfaces metric-embeddings







share|cite|improve this question











$endgroup$












  • $begingroup$
    Is there any example or counter example?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:53






  • 4




    $begingroup$
    Relevant: math.stackexchange.com/questions/1528046/…
    $endgroup$
    – Aknazar Kazhymurat
    Mar 20 at 8:57










  • $begingroup$
    So, is this an open problem?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:59






  • 3




    $begingroup$
    You might be interested in this question and its answers.
    $endgroup$
    – Michael Albanese
    Mar 20 at 12:00
















5












$begingroup$


Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $mathbb{R}^4?$






dg.differential-geometry riemannian-geometry riemann-surfaces metric-embeddings







share|cite|improve this question











$endgroup$












  • $begingroup$
    Is there any example or counter example?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:53






  • 4




    $begingroup$
    Relevant: math.stackexchange.com/questions/1528046/…
    $endgroup$
    – Aknazar Kazhymurat
    Mar 20 at 8:57










  • $begingroup$
    So, is this an open problem?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:59






  • 3




    $begingroup$
    You might be interested in this question and its answers.
    $endgroup$
    – Michael Albanese
    Mar 20 at 12:00














5












5








5


1



$begingroup$


Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $mathbb{R}^4?$






dg.differential-geometry riemannian-geometry riemann-surfaces metric-embeddings







share|cite|improve this question











$endgroup$




Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $mathbb{R}^4?$






dg.differential-geometry riemannian-geometry riemann-surfaces metric-embeddings




dg.differential-geometry riemannian-geometry riemann-surfaces metric-embeddings






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 20 at 8:57









Sean Lawton

3,97622248




3,97622248










asked Mar 20 at 8:49









GAUTAM NEELAKANTAN MEMANAGAUTAM NEELAKANTAN MEMANA

264




264












  • $begingroup$
    Is there any example or counter example?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:53






  • 4




    $begingroup$
    Relevant: math.stackexchange.com/questions/1528046/…
    $endgroup$
    – Aknazar Kazhymurat
    Mar 20 at 8:57










  • $begingroup$
    So, is this an open problem?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:59






  • 3




    $begingroup$
    You might be interested in this question and its answers.
    $endgroup$
    – Michael Albanese
    Mar 20 at 12:00


















  • $begingroup$
    Is there any example or counter example?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:53






  • 4




    $begingroup$
    Relevant: math.stackexchange.com/questions/1528046/…
    $endgroup$
    – Aknazar Kazhymurat
    Mar 20 at 8:57










  • $begingroup$
    So, is this an open problem?
    $endgroup$
    – GAUTAM NEELAKANTAN MEMANA
    Mar 20 at 8:59






  • 3




    $begingroup$
    You might be interested in this question and its answers.
    $endgroup$
    – Michael Albanese
    Mar 20 at 12:00
















$begingroup$
Is there any example or counter example?
$endgroup$
– GAUTAM NEELAKANTAN MEMANA
Mar 20 at 8:53




$begingroup$
Is there any example or counter example?
$endgroup$
– GAUTAM NEELAKANTAN MEMANA
Mar 20 at 8:53




4




4




$begingroup$
Relevant: math.stackexchange.com/questions/1528046/…
$endgroup$
– Aknazar Kazhymurat
Mar 20 at 8:57




$begingroup$
Relevant: math.stackexchange.com/questions/1528046/…
$endgroup$
– Aknazar Kazhymurat
Mar 20 at 8:57












$begingroup$
So, is this an open problem?
$endgroup$
– GAUTAM NEELAKANTAN MEMANA
Mar 20 at 8:59




$begingroup$
So, is this an open problem?
$endgroup$
– GAUTAM NEELAKANTAN MEMANA
Mar 20 at 8:59




3




3




$begingroup$
You might be interested in this question and its answers.
$endgroup$
– Michael Albanese
Mar 20 at 12:00




$begingroup$
You might be interested in this question and its answers.
$endgroup$
– Michael Albanese
Mar 20 at 12:00










2 Answers
2






active

oldest

votes


















11












$begingroup$

The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$. However, I believe the actual visualization problem for $ggeq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$-embedding of the flat torus, which has a $C^infty$-isometric embedding into $mathbb{R}^4$).



Since the smallest known $C^infty$-embedding for the hyperbolic plane is $mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open). Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$-embedded into $mathbb{R}^3$ for $rgeq 2$. Later Efimov generalized this to closed hyperbolic surfaces.



I believe these facts and references may be found in:



Isometric Embedding of Riemannian Manifolds in Euclidean Spaces by Qing Han, and Jia-Xing Hong.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
    $endgroup$
    – John Pardon
    Mar 20 at 14:30










  • $begingroup$
    @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
    $endgroup$
    – Sean Lawton
    Mar 20 at 14:33








  • 1




    $begingroup$
    That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
    $endgroup$
    – John Pardon
    Mar 20 at 14:38






  • 1




    $begingroup$
    There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
    $endgroup$
    – John Pardon
    Mar 20 at 14:45








  • 1




    $begingroup$
    @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
    $endgroup$
    – Sean Lawton
    Mar 20 at 14:48



















5












$begingroup$

I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{infty}$ isometric embedding $V to mathbb{R}^5$. See Gromov's Partial Differential Relations, pages 298 - 303.






share|cite|improve this answer











$endgroup$














    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: "504"
    };
    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',
    autoActivateHeartbeat: false,
    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%2fmathoverflow.net%2fquestions%2f325842%2fisometric-embedding-of-a-genus-g-surface%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    11












    $begingroup$

    The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$. However, I believe the actual visualization problem for $ggeq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$-embedding of the flat torus, which has a $C^infty$-isometric embedding into $mathbb{R}^4$).



    Since the smallest known $C^infty$-embedding for the hyperbolic plane is $mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open). Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$-embedded into $mathbb{R}^3$ for $rgeq 2$. Later Efimov generalized this to closed hyperbolic surfaces.



    I believe these facts and references may be found in:



    Isometric Embedding of Riemannian Manifolds in Euclidean Spaces by Qing Han, and Jia-Xing Hong.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
      $endgroup$
      – John Pardon
      Mar 20 at 14:30










    • $begingroup$
      @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:33








    • 1




      $begingroup$
      That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
      $endgroup$
      – John Pardon
      Mar 20 at 14:38






    • 1




      $begingroup$
      There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
      $endgroup$
      – John Pardon
      Mar 20 at 14:45








    • 1




      $begingroup$
      @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:48
















    11












    $begingroup$

    The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$. However, I believe the actual visualization problem for $ggeq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$-embedding of the flat torus, which has a $C^infty$-isometric embedding into $mathbb{R}^4$).



    Since the smallest known $C^infty$-embedding for the hyperbolic plane is $mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open). Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$-embedded into $mathbb{R}^3$ for $rgeq 2$. Later Efimov generalized this to closed hyperbolic surfaces.



    I believe these facts and references may be found in:



    Isometric Embedding of Riemannian Manifolds in Euclidean Spaces by Qing Han, and Jia-Xing Hong.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
      $endgroup$
      – John Pardon
      Mar 20 at 14:30










    • $begingroup$
      @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:33








    • 1




      $begingroup$
      That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
      $endgroup$
      – John Pardon
      Mar 20 at 14:38






    • 1




      $begingroup$
      There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
      $endgroup$
      – John Pardon
      Mar 20 at 14:45








    • 1




      $begingroup$
      @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:48














    11












    11








    11





    $begingroup$

    The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$. However, I believe the actual visualization problem for $ggeq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$-embedding of the flat torus, which has a $C^infty$-isometric embedding into $mathbb{R}^4$).



    Since the smallest known $C^infty$-embedding for the hyperbolic plane is $mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open). Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$-embedded into $mathbb{R}^3$ for $rgeq 2$. Later Efimov generalized this to closed hyperbolic surfaces.



    I believe these facts and references may be found in:



    Isometric Embedding of Riemannian Manifolds in Euclidean Spaces by Qing Han, and Jia-Xing Hong.






    share|cite|improve this answer











    $endgroup$



    The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$. However, I believe the actual visualization problem for $ggeq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$-embedding of the flat torus, which has a $C^infty$-isometric embedding into $mathbb{R}^4$).



    Since the smallest known $C^infty$-embedding for the hyperbolic plane is $mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open). Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$-embedded into $mathbb{R}^3$ for $rgeq 2$. Later Efimov generalized this to closed hyperbolic surfaces.



    I believe these facts and references may be found in:



    Isometric Embedding of Riemannian Manifolds in Euclidean Spaces by Qing Han, and Jia-Xing Hong.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Mar 20 at 14:42

























    answered Mar 20 at 9:08









    Sean LawtonSean Lawton

    3,97622248




    3,97622248








    • 1




      $begingroup$
      There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
      $endgroup$
      – John Pardon
      Mar 20 at 14:30










    • $begingroup$
      @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:33








    • 1




      $begingroup$
      That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
      $endgroup$
      – John Pardon
      Mar 20 at 14:38






    • 1




      $begingroup$
      There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
      $endgroup$
      – John Pardon
      Mar 20 at 14:45








    • 1




      $begingroup$
      @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:48














    • 1




      $begingroup$
      There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
      $endgroup$
      – John Pardon
      Mar 20 at 14:30










    • $begingroup$
      @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:33








    • 1




      $begingroup$
      That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
      $endgroup$
      – John Pardon
      Mar 20 at 14:38






    • 1




      $begingroup$
      There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
      $endgroup$
      – John Pardon
      Mar 20 at 14:45








    • 1




      $begingroup$
      @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
      $endgroup$
      – Sean Lawton
      Mar 20 at 14:48








    1




    1




    $begingroup$
    There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
    $endgroup$
    – John Pardon
    Mar 20 at 14:30




    $begingroup$
    There are visualizations of the Nash--Kuiper embeddings here: hevea-project.fr
    $endgroup$
    – John Pardon
    Mar 20 at 14:30












    $begingroup$
    @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
    $endgroup$
    – Sean Lawton
    Mar 20 at 14:33






    $begingroup$
    @JohnPardon Thanks for the link. However, I already knew about the flat tori and Nash spheres from Hevea; I referenced one in my answer. It is the hyperbolic ones for closed surfaces of $ggeq 2$ that are not presently visualized (as far as I know).
    $endgroup$
    – Sean Lawton
    Mar 20 at 14:33






    1




    1




    $begingroup$
    That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
    $endgroup$
    – John Pardon
    Mar 20 at 14:38




    $begingroup$
    That's a confusing place to put the link: the smooth flat embedding of the torus in R^4 and the C^1 flat embedding of the torus in R^3 aren't related at all.
    $endgroup$
    – John Pardon
    Mar 20 at 14:38




    1




    1




    $begingroup$
    There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
    $endgroup$
    – John Pardon
    Mar 20 at 14:45






    $begingroup$
    There isn't really any difference between the $g=1$ and $ggeq 2$ cases for the Nash--Kuiper construction of C^1 isometric embeddings; their method works for an arbitrary metric without regard to its curvature. I'm fairly certain that if one can render a flat torus in R^3, then one can render any surface with any metric in R^3, though the people at Hévéa might know better. (+1 btw)
    $endgroup$
    – John Pardon
    Mar 20 at 14:45






    1




    1




    $begingroup$
    @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
    $endgroup$
    – Sean Lawton
    Mar 20 at 14:48




    $begingroup$
    @JohnPardon Sure, I know the method works theoretically in general, but actually doing it is computationally complicated (which is why it hasn't been done).
    $endgroup$
    – Sean Lawton
    Mar 20 at 14:48











    5












    $begingroup$

    I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{infty}$ isometric embedding $V to mathbb{R}^5$. See Gromov's Partial Differential Relations, pages 298 - 303.






    share|cite|improve this answer











    $endgroup$


















      5












      $begingroup$

      I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{infty}$ isometric embedding $V to mathbb{R}^5$. See Gromov's Partial Differential Relations, pages 298 - 303.






      share|cite|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{infty}$ isometric embedding $V to mathbb{R}^5$. See Gromov's Partial Differential Relations, pages 298 - 303.






        share|cite|improve this answer











        $endgroup$



        I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{infty}$ isometric embedding $V to mathbb{R}^5$. See Gromov's Partial Differential Relations, pages 298 - 303.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 20 at 11:50

























        answered Mar 20 at 11:38









        Michael AlbaneseMichael Albanese

        7,88155393




        7,88155393






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to MathOverflow!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f325842%2fisometric-embedding-of-a-genus-g-surface%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

            How to change which sound is reproduced for terminal bell?

            Image
            3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
            Read more

            Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

            Image
            up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
            Read more

            Can I use Tabulator js library in my java Spring + Thymeleaf project?

            Image
            0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
            Read more