A question concerning the developing map of (G,X) manifolds












2












$begingroup$


Let $M$ be a $(G,X)$ manifold, that is we have local charts $(U,varphi_U)$ on $M$ with $varphi_U$ a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$.



Let $mathfrak{p}:widetilde{M}rightarrow M$ be the universal covering of $M$.



The developing map theorem introduces a local diffeomorphism $dev:widetilde{M}rightarrow X$.



Does the developing map locally commute with the (restricted) covering and local charts, i.e. for $widetilde{U}$ (small enough) $dev|_widetilde{U}=varphi_{U}circmathfrak{p}|_widetilde{U}$ for some $varphi_U$?






dg.differential-geometry







share|cite|improve this question







New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    It's hard to make sense of the question (the last sentence). What's "small enough $tilde{U}$"? what's "for some $varphi_U$" when $varphi_U$ is already introduced?
    $endgroup$
    – YCor
    4 hours ago
















2












$begingroup$


Let $M$ be a $(G,X)$ manifold, that is we have local charts $(U,varphi_U)$ on $M$ with $varphi_U$ a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$.



Let $mathfrak{p}:widetilde{M}rightarrow M$ be the universal covering of $M$.



The developing map theorem introduces a local diffeomorphism $dev:widetilde{M}rightarrow X$.



Does the developing map locally commute with the (restricted) covering and local charts, i.e. for $widetilde{U}$ (small enough) $dev|_widetilde{U}=varphi_{U}circmathfrak{p}|_widetilde{U}$ for some $varphi_U$?






dg.differential-geometry







share|cite|improve this question







New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    It's hard to make sense of the question (the last sentence). What's "small enough $tilde{U}$"? what's "for some $varphi_U$" when $varphi_U$ is already introduced?
    $endgroup$
    – YCor
    4 hours ago














2












2








2





$begingroup$


Let $M$ be a $(G,X)$ manifold, that is we have local charts $(U,varphi_U)$ on $M$ with $varphi_U$ a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$.



Let $mathfrak{p}:widetilde{M}rightarrow M$ be the universal covering of $M$.



The developing map theorem introduces a local diffeomorphism $dev:widetilde{M}rightarrow X$.



Does the developing map locally commute with the (restricted) covering and local charts, i.e. for $widetilde{U}$ (small enough) $dev|_widetilde{U}=varphi_{U}circmathfrak{p}|_widetilde{U}$ for some $varphi_U$?






dg.differential-geometry







share|cite|improve this question







New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




Let $M$ be a $(G,X)$ manifold, that is we have local charts $(U,varphi_U)$ on $M$ with $varphi_U$ a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$.



Let $mathfrak{p}:widetilde{M}rightarrow M$ be the universal covering of $M$.



The developing map theorem introduces a local diffeomorphism $dev:widetilde{M}rightarrow X$.



Does the developing map locally commute with the (restricted) covering and local charts, i.e. for $widetilde{U}$ (small enough) $dev|_widetilde{U}=varphi_{U}circmathfrak{p}|_widetilde{U}$ for some $varphi_U$?






dg.differential-geometry




dg.differential-geometry






share|cite|improve this question







New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 5 hours ago









user135350user135350

111




111




New contributor




user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






user135350 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    $begingroup$
    It's hard to make sense of the question (the last sentence). What's "small enough $tilde{U}$"? what's "for some $varphi_U$" when $varphi_U$ is already introduced?
    $endgroup$
    – YCor
    4 hours ago














  • 1




    $begingroup$
    It's hard to make sense of the question (the last sentence). What's "small enough $tilde{U}$"? what's "for some $varphi_U$" when $varphi_U$ is already introduced?
    $endgroup$
    – YCor
    4 hours ago








1




1




$begingroup$
It's hard to make sense of the question (the last sentence). What's "small enough $tilde{U}$"? what's "for some $varphi_U$" when $varphi_U$ is already introduced?
$endgroup$
– YCor
4 hours ago




$begingroup$
It's hard to make sense of the question (the last sentence). What's "small enough $tilde{U}$"? what's "for some $varphi_U$" when $varphi_U$ is already introduced?
$endgroup$
– YCor
4 hours ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

The best way to see this is to read the proof of the existence of the developing map.
The $(X,G)$ structure on $M$ induces an $(X,G)$-structure on $tilde M$ its universal cover such that for any $tilde xin tilde M$, write $x=p(tilde x)$ where $p$ is the covering map. Consider a chart $xin U$ such that there exists $tilde U$ which contains $tilde x$ such that the restriction of $p$ to $tilde U$, $p_{tilde U}:tilde Urightarrow U$ is an homeomorphism $phi_{tilde U}=phi_Ucirc p_{tilde U}$.



To construct the developing map, one fixes $tilde x_0in tilde M$ for every $tilde xin hat M$, one considers a path $c:[0,1]rightarrow hat M$ such that $c(0)=tilde x_0$ and $c(1)=tilde x$, then one consider chart $(tilde U_0,tilde phi_0),...(tilde U_n,tildephi_n)$ such that there exists a subdivision $[0=t_0,t_1,...,t_n=1]$ such that $c[t_i,t_{i+1}]subset tilde U_i$ and one sets:



$D(tilde x)=tilde g_0...g_{n-1}tildephi_n(tilde x)$ where $g_iin G$ such that $tilde phi_icirc{tildephi_{i+1}}^{-1}$ is the restriction of $g_i$ to $tildephi_{i+1}(tilde U_{i+1})$.



This shows that if we set $U_n=p(tilde U_n)$, we can define the chart $(U_n,g_0...g_{n-1}phi_n)$ of $x$ and the chart $(tilde U_n,g_0...g_{n-1}tilde phi_n)$ which answer your question.






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
    });


    }
    });






    user135350 is a new contributor. Be nice, and check out our Code of Conduct.










    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f322330%2fa-question-concerning-the-developing-map-of-g-x-manifolds%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









    2












    $begingroup$

    The best way to see this is to read the proof of the existence of the developing map.
    The $(X,G)$ structure on $M$ induces an $(X,G)$-structure on $tilde M$ its universal cover such that for any $tilde xin tilde M$, write $x=p(tilde x)$ where $p$ is the covering map. Consider a chart $xin U$ such that there exists $tilde U$ which contains $tilde x$ such that the restriction of $p$ to $tilde U$, $p_{tilde U}:tilde Urightarrow U$ is an homeomorphism $phi_{tilde U}=phi_Ucirc p_{tilde U}$.



    To construct the developing map, one fixes $tilde x_0in tilde M$ for every $tilde xin hat M$, one considers a path $c:[0,1]rightarrow hat M$ such that $c(0)=tilde x_0$ and $c(1)=tilde x$, then one consider chart $(tilde U_0,tilde phi_0),...(tilde U_n,tildephi_n)$ such that there exists a subdivision $[0=t_0,t_1,...,t_n=1]$ such that $c[t_i,t_{i+1}]subset tilde U_i$ and one sets:



    $D(tilde x)=tilde g_0...g_{n-1}tildephi_n(tilde x)$ where $g_iin G$ such that $tilde phi_icirc{tildephi_{i+1}}^{-1}$ is the restriction of $g_i$ to $tildephi_{i+1}(tilde U_{i+1})$.



    This shows that if we set $U_n=p(tilde U_n)$, we can define the chart $(U_n,g_0...g_{n-1}phi_n)$ of $x$ and the chart $(tilde U_n,g_0...g_{n-1}tilde phi_n)$ which answer your question.






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      The best way to see this is to read the proof of the existence of the developing map.
      The $(X,G)$ structure on $M$ induces an $(X,G)$-structure on $tilde M$ its universal cover such that for any $tilde xin tilde M$, write $x=p(tilde x)$ where $p$ is the covering map. Consider a chart $xin U$ such that there exists $tilde U$ which contains $tilde x$ such that the restriction of $p$ to $tilde U$, $p_{tilde U}:tilde Urightarrow U$ is an homeomorphism $phi_{tilde U}=phi_Ucirc p_{tilde U}$.



      To construct the developing map, one fixes $tilde x_0in tilde M$ for every $tilde xin hat M$, one considers a path $c:[0,1]rightarrow hat M$ such that $c(0)=tilde x_0$ and $c(1)=tilde x$, then one consider chart $(tilde U_0,tilde phi_0),...(tilde U_n,tildephi_n)$ such that there exists a subdivision $[0=t_0,t_1,...,t_n=1]$ such that $c[t_i,t_{i+1}]subset tilde U_i$ and one sets:



      $D(tilde x)=tilde g_0...g_{n-1}tildephi_n(tilde x)$ where $g_iin G$ such that $tilde phi_icirc{tildephi_{i+1}}^{-1}$ is the restriction of $g_i$ to $tildephi_{i+1}(tilde U_{i+1})$.



      This shows that if we set $U_n=p(tilde U_n)$, we can define the chart $(U_n,g_0...g_{n-1}phi_n)$ of $x$ and the chart $(tilde U_n,g_0...g_{n-1}tilde phi_n)$ which answer your question.






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        The best way to see this is to read the proof of the existence of the developing map.
        The $(X,G)$ structure on $M$ induces an $(X,G)$-structure on $tilde M$ its universal cover such that for any $tilde xin tilde M$, write $x=p(tilde x)$ where $p$ is the covering map. Consider a chart $xin U$ such that there exists $tilde U$ which contains $tilde x$ such that the restriction of $p$ to $tilde U$, $p_{tilde U}:tilde Urightarrow U$ is an homeomorphism $phi_{tilde U}=phi_Ucirc p_{tilde U}$.



        To construct the developing map, one fixes $tilde x_0in tilde M$ for every $tilde xin hat M$, one considers a path $c:[0,1]rightarrow hat M$ such that $c(0)=tilde x_0$ and $c(1)=tilde x$, then one consider chart $(tilde U_0,tilde phi_0),...(tilde U_n,tildephi_n)$ such that there exists a subdivision $[0=t_0,t_1,...,t_n=1]$ such that $c[t_i,t_{i+1}]subset tilde U_i$ and one sets:



        $D(tilde x)=tilde g_0...g_{n-1}tildephi_n(tilde x)$ where $g_iin G$ such that $tilde phi_icirc{tildephi_{i+1}}^{-1}$ is the restriction of $g_i$ to $tildephi_{i+1}(tilde U_{i+1})$.



        This shows that if we set $U_n=p(tilde U_n)$, we can define the chart $(U_n,g_0...g_{n-1}phi_n)$ of $x$ and the chart $(tilde U_n,g_0...g_{n-1}tilde phi_n)$ which answer your question.






        share|cite|improve this answer











        $endgroup$



        The best way to see this is to read the proof of the existence of the developing map.
        The $(X,G)$ structure on $M$ induces an $(X,G)$-structure on $tilde M$ its universal cover such that for any $tilde xin tilde M$, write $x=p(tilde x)$ where $p$ is the covering map. Consider a chart $xin U$ such that there exists $tilde U$ which contains $tilde x$ such that the restriction of $p$ to $tilde U$, $p_{tilde U}:tilde Urightarrow U$ is an homeomorphism $phi_{tilde U}=phi_Ucirc p_{tilde U}$.



        To construct the developing map, one fixes $tilde x_0in tilde M$ for every $tilde xin hat M$, one considers a path $c:[0,1]rightarrow hat M$ such that $c(0)=tilde x_0$ and $c(1)=tilde x$, then one consider chart $(tilde U_0,tilde phi_0),...(tilde U_n,tildephi_n)$ such that there exists a subdivision $[0=t_0,t_1,...,t_n=1]$ such that $c[t_i,t_{i+1}]subset tilde U_i$ and one sets:



        $D(tilde x)=tilde g_0...g_{n-1}tildephi_n(tilde x)$ where $g_iin G$ such that $tilde phi_icirc{tildephi_{i+1}}^{-1}$ is the restriction of $g_i$ to $tildephi_{i+1}(tilde U_{i+1})$.



        This shows that if we set $U_n=p(tilde U_n)$, we can define the chart $(U_n,g_0...g_{n-1}phi_n)$ of $x$ and the chart $(tilde U_n,g_0...g_{n-1}tilde phi_n)$ which answer your question.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 4 hours ago

























        answered 4 hours ago









        Tsemo AristideTsemo Aristide

        2,6621616




        2,6621616






















            user135350 is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            user135350 is a new contributor. Be nice, and check out our Code of Conduct.













            user135350 is a new contributor. Be nice, and check out our Code of Conduct.












            user135350 is a new contributor. Be nice, and check out our Code of Conduct.
















            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%2f322330%2fa-question-concerning-the-developing-map-of-g-x-manifolds%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

            Knooppunt Holsloot

            Image
            Knooppunt Holsloot Kenmerken Wegen A37 × N34 Type knooppunt klaverblad Portaal     Verkeer & Vervoer Knooppunt Holsloot is een verkeersknooppunt op de kruising van de autosnelweg A37 en de N34 in Drenthe, tussen Emmen en Coevorden. Oorspronkelijk was deze aansluiting een afritconstructie van het type Haarlemmermeeraansluiting, zoals bij Knooppunt Hooipolder. Sinds 2002 is het knooppunt omgebouwd tot een volledig klaverblad. Als noviteit werd hier voor het eerst gebruikgemaakt van dimbare verlichting in plaats van de normale vaste tijden waarop de verlichting aan en uit gaat, wat een grote kostenbesparing opleverde. [1] Dat Holsloot als volwaardig knooppunt is uitgevoerd, waarbij alle verkeersstromen en afbuigingen ononderbroken kunnen doorgaan, is uitzonderlijk. Het betreft hier namelijk een aansluiting tussen een autosnelweg en niet-snelweg. Een vergelijkbare situatie treft men nabij Hoensbroek bij het knooppunt Ten Esschen.     Aa
            Read more

            Altaar (religie)

            Image
            Deel van de serie over kerkelijk gerei waaronder paramenten en liturgisch vaatwerk gebruikt in de liturgie Liturgisch vaatwerk Vasa sacra Miskelk · Pateen Kelklepeltje Ciborie · Monstrans Pyxis · Custodiale Vasa non sacra Ampullen · Wijwatervat Olievaatje Wierookvat · Ablutievat Paramenten Amict · Albe · Baarkleed · Cingel Tuniek · Dalmatiek · Fanon Kazuifel · Manipel · Stola Gremiale · Benedictievelum Mijter Koorkledij Rochet · Superplie Koorkap · Cappa magna Kovel Kelkgerei Bursa · Kelkvelum Ciborievelum Kerklinnen Corporale · Kelkdoekje Lavabodoekje · Palla Altaardwaal Kerkinterieur Altaar · Ambo Biechtstoel · Communiebank Doksaal · Doopvont Faldistorium Godslamp · Hoogaltaar Heilig Kruisaltaar Katheder · Preekstoel Sedilia Tabernakel · Volksaltaar Liturgische boeken Altaarmissaal Benedictionale · Brevier Evangeliarium · Evangelistarium Graduale · Kyriale Lectionarium Psalter · Rituaal Sacramentarium · Volksmissaal Overige Flambou
            Read more

            Gregoriusmis

            Image
            Een afbeelding van de Gregoriusmis uit een getijdenboek gemaakt in de zuidelijke Nederlanden tussen 1500 en 1525 Gregoriusmis, Meester van het Akense Altaar, ca. 1500-1525. Collectie Museum Catharijneconvent De Gregoriusmis of de Mis van Sint Gregorius is een motief in katholieke kunst, dat het eerst verscheen in de late Middeleeuwen en dat nog steeds werd gebruikt tijdens de contrareformatie. Paus Gregorius I (ca. 540-604) is afgebeeld, de H. Mis opdragend, terwijl Christus als de Man van Smarten verschijnt op het altaar dat voor Sint Gregorius staat. Volgens de legende had St. Gregorius gebeden om een teken voor een betwijfelaar van de transsubstantiatieleer, waarop de lijdende Christus tijdens het opdragen van de Mis verscheen om hem te overtuigen en waarbij zijn bloed in de altaarkelk liep. Vaak zijn ook de passiewerktuigen afgebeeld. Zie de categorie Mass of Saint Gregory van Wikimedia Commons voor mediabestanden over dit onderwerp. This page is onl
            Read more