Is every local diffeomorphism from $U subseteq mathbb{R}^k$ to a $V subseteq M$ equivalent to a chart?Smooth...

Signed and unsigned numbers

How can I ensure my trip to the UK will not have to be cancelled because of Brexit?

How to draw cubes in a 3 dimensional plane

Does this video of collapsing warehouse shelves show a real incident?

If I receive an SOS signal, what is the proper response?

What is the magic ball of every day?

Is it "Vierergruppe" or "Viergruppe", or is there a distinction?

NASA's RS-25 Engines shut down time

How to write ı (i without dot) character in pgf-pie

Vocabulary for giving just numbers, not a full answer

Are there historical instances of the capital of a colonising country being temporarily or permanently shifted to one of its colonies?

Latex does not go to next line

Plausibility of Mushroom Buildings

Why was Goose renamed from Chewie for the Captain Marvel film?

Bash script should only kill those instances of another script's that it has launched

meaning and function of 幸 in "则幸分我一杯羹"

How do I express some one as a black person?

When traveling to Europe from North America, do I need to purchase a different power strip?

Motivation for Zeta Function of an Algebraic Variety

Good for you! in Russian

PTIJ: wiping amalek’s memory?

They call me Inspector Morse

What problems would a superhuman have whose skin is constantly hot?

How can I get players to stop ignoring or overlooking the plot hooks I'm giving them?



Is every local diffeomorphism from $U subseteq mathbb{R}^k$ to a $V subseteq M$ equivalent to a chart?


Smooth chart in what sense?What does “smoothly compatible” mean?Non-smooth internal function of composition which happens to be smoothIs every local diffeomorphic chart a local isometry?Smooth structure of embedded submanifoldssmooth submersions and maps with local sectionsBijective local diffeomorphism is a diffeomorphismSmooth map between vector bundles from local coordinatesEquivalence of the two definitions of smooth mapsChoosing Coordinate Charts — Smooth Functions on Manifolds













0












$begingroup$


If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?



It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.






differential-geometry smooth-manifolds







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?



    It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.






    differential-geometry smooth-manifolds







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?



      It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.






      differential-geometry smooth-manifolds







      share|cite|improve this question











      $endgroup$




      If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?



      It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.






      differential-geometry smooth-manifolds




      differential-geometry smooth-manifolds






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago







      Jeff

















      asked 2 days ago









      JeffJeff

      13318




      13318






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).



          Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
          phi circ Fvert_{W cap X}^{-1})$           is a chart for $M$.



          In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.






          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%2f3140788%2fis-every-local-diffeomorphism-from-u-subseteq-mathbbrk-to-a-v-subseteq%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












            $begingroup$

            Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).



            Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
            phi circ Fvert_{W cap X}^{-1})$             is a chart for $M$.



            In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.






            share|cite|improve this answer











            $endgroup$


















              2












              $begingroup$

              Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).



              Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
              phi circ Fvert_{W cap X}^{-1})$               is a chart for $M$.



              In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.






              share|cite|improve this answer











              $endgroup$
















                2












                2








                2





                $begingroup$

                Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).



                Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
                phi circ Fvert_{W cap X}^{-1})$                 is a chart for $M$.



                In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.






                share|cite|improve this answer











                $endgroup$



                Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).



                Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
                phi circ Fvert_{W cap X}^{-1})$                 is a chart for $M$.



                In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 2 days ago

























                answered 2 days ago









                setnosetsetnoset

                36518




                36518






























                    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%2f3140788%2fis-every-local-diffeomorphism-from-u-subseteq-mathbbrk-to-a-v-subseteq%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

                    Nidaros erkebispedøme

                    Image
                    For det lutherske bispedømet, sjå Nidaros bispedøme. Nidaros erkebispedøme var eit katolsk erkebispedøme med sete i Nidaros. Det vart oppretta av kong Olav Kyrre mellom 1070 og 1080 som bispedøme. Bispedømet vart opphøgd til erkebispedøme i 1153. Erkebispedømet vart innstifta av pave Anastasius IV i 1154, og var i funksjon fram til reformasjonen i 1537. Erkebiskopen sitt hovudsete var Erkebispegarden, rett ved domkyrkja Nidarosdomen. Frå 1152 til 1154 var Nicholas Breakspear i Skandinavia som paveleg utsending, og organiserte opprettinga av erkesetet. Innhaldsliste 1 Utbreiing og jurisdiksjon 2 Overhyrdar 2.1 Lydbiskopar 3 Biskopar 3.1 Biskopar 3.2 Erkebiskopar Utbreiing og jurisdiksjon | Kart over den nordiske kyrkjeprovinsen (1153-1387) Erkebispedømet omfatta i tillegg til Noreg øyane i vest i Noregsveldet. Nidaros var eit metropolitanerkebispedøme med fylgjande underliggjande bispedøme suffraganar: Bj...
                    Read more

                    Birsay

                    Image
                    Tettstaden Birsay på Orknøyane. Ruinane av jarleplasset om lag midt i biletet. (Fotografi frå 2012) Utsikt mot Birsay og Brough of Brisay (fotografi frå 2009) Birsay er ein tettstad i eit sokn (parish) med same namn, nordvest på øya Mainland [Note 1] på Orknøyane, om lag 31 km frå Kirkwall. Staden er omtala fleire stader i Orknøyingasoga [1] , der han er kalla «Byrgesherad» (norrønt: Birgisherað ). Namnet skriv seg truleg frå elvenamnet Birgisá [2] . Innhaldsliste 1 Den gamle hovudstaden 2 Bispesetet 3 Jarlepalasset 4 Kjelder og utfyllande opplysningar 4.1 Notar 4.2 Fotnotar 4.3 Bakgrunnsstoff Den gamle hovudstaden | Hovudartikkelen for dette emnet er Torfinn Sigurdsson . Soknet er i dag i hovudsak ei jordbrukskommune men er mest kjend for dei mange kulturminna i området. Tettstaden Birsay blir gjerne omtala som «den gamle hovudstaden på Orknøyane» [3] , og ein viser d...
                    Read more

                    Was Woodrow Wilson really a Liberal?Was World War I a war of liberals against authoritarians?Founding Fathers...

                    Image
                    Do VLANs within a subnet need to have their own subnet for router on a stick? What are the differences between the usage of 'it' and 'they'? Dragon forelimb placement How do we improve the relationship with a client software team that performs poorly and is becoming less collaborative? What typically incentivizes a professor to change jobs to a lower ranking university? How to write a macro that is braces sensitive? Modeling an IPv4 Address Writing rule stating superpower from different root cause is bad writing Can I ask the recruiters in my resume to put the reason why I am rejected? Approximately how much travel t...
                    Read more