Computing Fubini-Study metric from the formal definition












1












$begingroup$



Definition: the Fubini-Study metric $g_{FB}$ on $mathbb{CP}^n$ is the only metric which makes the projection $pi:(mathbb{S}^{2n+1},g)to(mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is the standard metric)




I'm trying to deduce the coefficients of $g_{FB}$ in charts, i.e.:
begin{align*}
g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial x_beta}right)=g_{FB}left(frac{partial}{partial y_alpha},frac{partial}{partial y_beta}right)&=frac{1}{(1+|z|^2)^2}((1+|z|^2)delta_{alphabeta}-(x_alpha x_beta+y_alpha y_beta))\
g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial y_beta}right)&=frac{-1}{(1+|z|^2)^2}(x_alpha y_beta-y_alpha x_beta)
end{align*}



Since $x_alpha x_beta+y_alpha y_beta=text{Re}(z_alphaoverline{z}_beta)$ and $-(x_alpha y_beta-y_alpha x_beta)=text{Im}(z_alphaoverline{z}_beta)$, I thought maybe it would be a good idea to use complex variables, but I don't know how to do that from the condition that $pi$ is a Riemannian submersion.



Any suggestions?






riemannian-geometry complex-geometry projective-space







share|cite|improve this question











$endgroup$

















    1












    $begingroup$



    Definition: the Fubini-Study metric $g_{FB}$ on $mathbb{CP}^n$ is the only metric which makes the projection $pi:(mathbb{S}^{2n+1},g)to(mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is the standard metric)




    I'm trying to deduce the coefficients of $g_{FB}$ in charts, i.e.:
    begin{align*}
    g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial x_beta}right)=g_{FB}left(frac{partial}{partial y_alpha},frac{partial}{partial y_beta}right)&=frac{1}{(1+|z|^2)^2}((1+|z|^2)delta_{alphabeta}-(x_alpha x_beta+y_alpha y_beta))\
    g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial y_beta}right)&=frac{-1}{(1+|z|^2)^2}(x_alpha y_beta-y_alpha x_beta)
    end{align*}



    Since $x_alpha x_beta+y_alpha y_beta=text{Re}(z_alphaoverline{z}_beta)$ and $-(x_alpha y_beta-y_alpha x_beta)=text{Im}(z_alphaoverline{z}_beta)$, I thought maybe it would be a good idea to use complex variables, but I don't know how to do that from the condition that $pi$ is a Riemannian submersion.



    Any suggestions?






    riemannian-geometry complex-geometry projective-space







    share|cite|improve this question











    $endgroup$















      1












      1








      1


      1



      $begingroup$



      Definition: the Fubini-Study metric $g_{FB}$ on $mathbb{CP}^n$ is the only metric which makes the projection $pi:(mathbb{S}^{2n+1},g)to(mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is the standard metric)




      I'm trying to deduce the coefficients of $g_{FB}$ in charts, i.e.:
      begin{align*}
      g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial x_beta}right)=g_{FB}left(frac{partial}{partial y_alpha},frac{partial}{partial y_beta}right)&=frac{1}{(1+|z|^2)^2}((1+|z|^2)delta_{alphabeta}-(x_alpha x_beta+y_alpha y_beta))\
      g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial y_beta}right)&=frac{-1}{(1+|z|^2)^2}(x_alpha y_beta-y_alpha x_beta)
      end{align*}



      Since $x_alpha x_beta+y_alpha y_beta=text{Re}(z_alphaoverline{z}_beta)$ and $-(x_alpha y_beta-y_alpha x_beta)=text{Im}(z_alphaoverline{z}_beta)$, I thought maybe it would be a good idea to use complex variables, but I don't know how to do that from the condition that $pi$ is a Riemannian submersion.



      Any suggestions?






      riemannian-geometry complex-geometry projective-space







      share|cite|improve this question











      $endgroup$





      Definition: the Fubini-Study metric $g_{FB}$ on $mathbb{CP}^n$ is the only metric which makes the projection $pi:(mathbb{S}^{2n+1},g)to(mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is the standard metric)




      I'm trying to deduce the coefficients of $g_{FB}$ in charts, i.e.:
      begin{align*}
      g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial x_beta}right)=g_{FB}left(frac{partial}{partial y_alpha},frac{partial}{partial y_beta}right)&=frac{1}{(1+|z|^2)^2}((1+|z|^2)delta_{alphabeta}-(x_alpha x_beta+y_alpha y_beta))\
      g_{FB}left(frac{partial}{partial x_alpha},frac{partial}{partial y_beta}right)&=frac{-1}{(1+|z|^2)^2}(x_alpha y_beta-y_alpha x_beta)
      end{align*}



      Since $x_alpha x_beta+y_alpha y_beta=text{Re}(z_alphaoverline{z}_beta)$ and $-(x_alpha y_beta-y_alpha x_beta)=text{Im}(z_alphaoverline{z}_beta)$, I thought maybe it would be a good idea to use complex variables, but I don't know how to do that from the condition that $pi$ is a Riemannian submersion.



      Any suggestions?






      riemannian-geometry complex-geometry projective-space




      riemannian-geometry complex-geometry projective-space






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 13 '18 at 17:44







      rmdmc89

















      asked Dec 12 '18 at 20:50









      rmdmc89rmdmc89

      2,1221922




      2,1221922






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          It's indeed very natural to use the hermitian geometry structure here. Let $e_0;e_1,dots e_n$ be a unitary frame at the point $[Z]=[e_0]inBbb P^n$. Then
          $$de_0 = omega_{0bar 0}e_0 + sum omega_{0bar j}e_j$$
          and the hermitian metric on $Bbb P^n$ is given by $sum |omega_{0bar j}|^2 = sum omega_{0bar j}overline{omega_{0bar j}}$. In one of your charts, say $Z_0ne 0$, we take coordinates by setting $Z=(1,z)$ and $e_0 = Z/|Z|$.



          Now, note that $omega_{0bar 0} = i,dtheta$ where $e^{itheta}$ gives the fiber of your Riemannian submersion. Moreover, denoting the hermitian inner product by $(cdot,cdot)$,
          $$sum |omega_{0bar j}|^2 = (de_0,de_0) - |omega_{0bar 0}|^2 = (de_0,de_0)-|(de_0,e_0)|^2.$$
          Substituting $e_0 = dfrac{(1,z)}{|(1,z)|}$ and differentiating appropriately, you'll get your desired formula for the metric as a $2$-tensor in terms of $dz_j$ and $dbar z_j$.






          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: "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',
            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%2fmath.stackexchange.com%2fquestions%2f3037197%2fcomputing-fubini-study-metric-from-the-formal-definition%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









            1












            $begingroup$

            It's indeed very natural to use the hermitian geometry structure here. Let $e_0;e_1,dots e_n$ be a unitary frame at the point $[Z]=[e_0]inBbb P^n$. Then
            $$de_0 = omega_{0bar 0}e_0 + sum omega_{0bar j}e_j$$
            and the hermitian metric on $Bbb P^n$ is given by $sum |omega_{0bar j}|^2 = sum omega_{0bar j}overline{omega_{0bar j}}$. In one of your charts, say $Z_0ne 0$, we take coordinates by setting $Z=(1,z)$ and $e_0 = Z/|Z|$.



            Now, note that $omega_{0bar 0} = i,dtheta$ where $e^{itheta}$ gives the fiber of your Riemannian submersion. Moreover, denoting the hermitian inner product by $(cdot,cdot)$,
            $$sum |omega_{0bar j}|^2 = (de_0,de_0) - |omega_{0bar 0}|^2 = (de_0,de_0)-|(de_0,e_0)|^2.$$
            Substituting $e_0 = dfrac{(1,z)}{|(1,z)|}$ and differentiating appropriately, you'll get your desired formula for the metric as a $2$-tensor in terms of $dz_j$ and $dbar z_j$.






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              It's indeed very natural to use the hermitian geometry structure here. Let $e_0;e_1,dots e_n$ be a unitary frame at the point $[Z]=[e_0]inBbb P^n$. Then
              $$de_0 = omega_{0bar 0}e_0 + sum omega_{0bar j}e_j$$
              and the hermitian metric on $Bbb P^n$ is given by $sum |omega_{0bar j}|^2 = sum omega_{0bar j}overline{omega_{0bar j}}$. In one of your charts, say $Z_0ne 0$, we take coordinates by setting $Z=(1,z)$ and $e_0 = Z/|Z|$.



              Now, note that $omega_{0bar 0} = i,dtheta$ where $e^{itheta}$ gives the fiber of your Riemannian submersion. Moreover, denoting the hermitian inner product by $(cdot,cdot)$,
              $$sum |omega_{0bar j}|^2 = (de_0,de_0) - |omega_{0bar 0}|^2 = (de_0,de_0)-|(de_0,e_0)|^2.$$
              Substituting $e_0 = dfrac{(1,z)}{|(1,z)|}$ and differentiating appropriately, you'll get your desired formula for the metric as a $2$-tensor in terms of $dz_j$ and $dbar z_j$.






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                It's indeed very natural to use the hermitian geometry structure here. Let $e_0;e_1,dots e_n$ be a unitary frame at the point $[Z]=[e_0]inBbb P^n$. Then
                $$de_0 = omega_{0bar 0}e_0 + sum omega_{0bar j}e_j$$
                and the hermitian metric on $Bbb P^n$ is given by $sum |omega_{0bar j}|^2 = sum omega_{0bar j}overline{omega_{0bar j}}$. In one of your charts, say $Z_0ne 0$, we take coordinates by setting $Z=(1,z)$ and $e_0 = Z/|Z|$.



                Now, note that $omega_{0bar 0} = i,dtheta$ where $e^{itheta}$ gives the fiber of your Riemannian submersion. Moreover, denoting the hermitian inner product by $(cdot,cdot)$,
                $$sum |omega_{0bar j}|^2 = (de_0,de_0) - |omega_{0bar 0}|^2 = (de_0,de_0)-|(de_0,e_0)|^2.$$
                Substituting $e_0 = dfrac{(1,z)}{|(1,z)|}$ and differentiating appropriately, you'll get your desired formula for the metric as a $2$-tensor in terms of $dz_j$ and $dbar z_j$.






                share|cite|improve this answer











                $endgroup$



                It's indeed very natural to use the hermitian geometry structure here. Let $e_0;e_1,dots e_n$ be a unitary frame at the point $[Z]=[e_0]inBbb P^n$. Then
                $$de_0 = omega_{0bar 0}e_0 + sum omega_{0bar j}e_j$$
                and the hermitian metric on $Bbb P^n$ is given by $sum |omega_{0bar j}|^2 = sum omega_{0bar j}overline{omega_{0bar j}}$. In one of your charts, say $Z_0ne 0$, we take coordinates by setting $Z=(1,z)$ and $e_0 = Z/|Z|$.



                Now, note that $omega_{0bar 0} = i,dtheta$ where $e^{itheta}$ gives the fiber of your Riemannian submersion. Moreover, denoting the hermitian inner product by $(cdot,cdot)$,
                $$sum |omega_{0bar j}|^2 = (de_0,de_0) - |omega_{0bar 0}|^2 = (de_0,de_0)-|(de_0,e_0)|^2.$$
                Substituting $e_0 = dfrac{(1,z)}{|(1,z)|}$ and differentiating appropriately, you'll get your desired formula for the metric as a $2$-tensor in terms of $dz_j$ and $dbar z_j$.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 13 '18 at 23:31

























                answered Dec 13 '18 at 23:22









                Ted ShifrinTed Shifrin

                63.7k44591




                63.7k44591






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • 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%2fmath.stackexchange.com%2fquestions%2f3037197%2fcomputing-fubini-study-metric-from-the-formal-definition%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

                    Ellipse (mathématiques)

                    Image
                    Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
                    Read more

                    Quarter-circle Tiles

                    Image
                    up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
                    Read more

                    Mont Emei

                    Image
                    Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
                    Read more