What does $TM$, $T^*M$ and $*$ mean?(Definition Check)












0












$begingroup$


I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.



However, I encountered some notation confusion:




  1. What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?


  2. What's the difference between $TM$ and $T^*M$?


  3. Also, there seemed to be a custom to use $*$ for either space or functions, i.e. $V^*$ or $varphi_*$. What does $*$ indicate here? Does it indicate dual space or something else?



(The preview of the material is available on amazon. )



Image I:



enter image description here



Image II:



enter image description here










share|cite|improve this question









$endgroup$












  • $begingroup$
    $TM$ is the tangent bundle of the manifold $M$.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:18










  • $begingroup$
    @MattSamuel Thank you, so what does $*$ stands for?
    $endgroup$
    – user9976437
    Jan 6 at 22:22






  • 1




    $begingroup$
    $T^*M$ is the cotangent bundle; $varphi_*$ usually will denote an induced map on tangent bundle, homology, fundamental group, or some other derived object.
    $endgroup$
    – Danu
    Jan 6 at 22:23






  • 1




    $begingroup$
    On a vector space, a superscript $*$ usually indicates the dual.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:27










  • $begingroup$
    Why would someone downvote this question?
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:30
















0












$begingroup$


I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.



However, I encountered some notation confusion:




  1. What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?


  2. What's the difference between $TM$ and $T^*M$?


  3. Also, there seemed to be a custom to use $*$ for either space or functions, i.e. $V^*$ or $varphi_*$. What does $*$ indicate here? Does it indicate dual space or something else?



(The preview of the material is available on amazon. )



Image I:



enter image description here



Image II:



enter image description here










share|cite|improve this question









$endgroup$












  • $begingroup$
    $TM$ is the tangent bundle of the manifold $M$.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:18










  • $begingroup$
    @MattSamuel Thank you, so what does $*$ stands for?
    $endgroup$
    – user9976437
    Jan 6 at 22:22






  • 1




    $begingroup$
    $T^*M$ is the cotangent bundle; $varphi_*$ usually will denote an induced map on tangent bundle, homology, fundamental group, or some other derived object.
    $endgroup$
    – Danu
    Jan 6 at 22:23






  • 1




    $begingroup$
    On a vector space, a superscript $*$ usually indicates the dual.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:27










  • $begingroup$
    Why would someone downvote this question?
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:30














0












0








0





$begingroup$


I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.



However, I encountered some notation confusion:




  1. What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?


  2. What's the difference between $TM$ and $T^*M$?


  3. Also, there seemed to be a custom to use $*$ for either space or functions, i.e. $V^*$ or $varphi_*$. What does $*$ indicate here? Does it indicate dual space or something else?



(The preview of the material is available on amazon. )



Image I:



enter image description here



Image II:



enter image description here










share|cite|improve this question









$endgroup$




I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.



However, I encountered some notation confusion:




  1. What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?


  2. What's the difference between $TM$ and $T^*M$?


  3. Also, there seemed to be a custom to use $*$ for either space or functions, i.e. $V^*$ or $varphi_*$. What does $*$ indicate here? Does it indicate dual space or something else?



(The preview of the material is available on amazon. )



Image I:



enter image description here



Image II:



enter image description here







differential-geometry notation definition






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 6 at 22:15









user9976437user9976437

759




759












  • $begingroup$
    $TM$ is the tangent bundle of the manifold $M$.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:18










  • $begingroup$
    @MattSamuel Thank you, so what does $*$ stands for?
    $endgroup$
    – user9976437
    Jan 6 at 22:22






  • 1




    $begingroup$
    $T^*M$ is the cotangent bundle; $varphi_*$ usually will denote an induced map on tangent bundle, homology, fundamental group, or some other derived object.
    $endgroup$
    – Danu
    Jan 6 at 22:23






  • 1




    $begingroup$
    On a vector space, a superscript $*$ usually indicates the dual.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:27










  • $begingroup$
    Why would someone downvote this question?
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:30


















  • $begingroup$
    $TM$ is the tangent bundle of the manifold $M$.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:18










  • $begingroup$
    @MattSamuel Thank you, so what does $*$ stands for?
    $endgroup$
    – user9976437
    Jan 6 at 22:22






  • 1




    $begingroup$
    $T^*M$ is the cotangent bundle; $varphi_*$ usually will denote an induced map on tangent bundle, homology, fundamental group, or some other derived object.
    $endgroup$
    – Danu
    Jan 6 at 22:23






  • 1




    $begingroup$
    On a vector space, a superscript $*$ usually indicates the dual.
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:27










  • $begingroup$
    Why would someone downvote this question?
    $endgroup$
    – Matt Samuel
    Jan 6 at 22:30
















$begingroup$
$TM$ is the tangent bundle of the manifold $M$.
$endgroup$
– Matt Samuel
Jan 6 at 22:18




$begingroup$
$TM$ is the tangent bundle of the manifold $M$.
$endgroup$
– Matt Samuel
Jan 6 at 22:18












$begingroup$
@MattSamuel Thank you, so what does $*$ stands for?
$endgroup$
– user9976437
Jan 6 at 22:22




$begingroup$
@MattSamuel Thank you, so what does $*$ stands for?
$endgroup$
– user9976437
Jan 6 at 22:22




1




1




$begingroup$
$T^*M$ is the cotangent bundle; $varphi_*$ usually will denote an induced map on tangent bundle, homology, fundamental group, or some other derived object.
$endgroup$
– Danu
Jan 6 at 22:23




$begingroup$
$T^*M$ is the cotangent bundle; $varphi_*$ usually will denote an induced map on tangent bundle, homology, fundamental group, or some other derived object.
$endgroup$
– Danu
Jan 6 at 22:23




1




1




$begingroup$
On a vector space, a superscript $*$ usually indicates the dual.
$endgroup$
– Matt Samuel
Jan 6 at 22:27




$begingroup$
On a vector space, a superscript $*$ usually indicates the dual.
$endgroup$
– Matt Samuel
Jan 6 at 22:27












$begingroup$
Why would someone downvote this question?
$endgroup$
– Matt Samuel
Jan 6 at 22:30




$begingroup$
Why would someone downvote this question?
$endgroup$
– Matt Samuel
Jan 6 at 22:30










1 Answer
1






active

oldest

votes


















2












$begingroup$

$T_pM$ is the space of vectors tangent to $M$ at $p$, and $T_p^*M=(T_pM)^*$ is the corresponding dual space.



$TM$ is the tangent bundle of manifold, $T^*M$ is the cotangent bundle of the manifold.
$$ TM=bigsqcup_{pin M} T_pM$$
subject to the "gluing" conditions that $TM$ is "locally" of the form $Utimes mathbb{R}^n$ for $U$ an open neighborhood of $M$. These objects are called smooth vector bundles and you can read more about them in a book like



$(1)$ Lee's Introduction to Smooth Manifolds



$(2)$ Tu's An Introduction to Manifolds.






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%2f3064443%2fwhat-does-tm-tm-and-meandefinition-check%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












    $begingroup$

    $T_pM$ is the space of vectors tangent to $M$ at $p$, and $T_p^*M=(T_pM)^*$ is the corresponding dual space.



    $TM$ is the tangent bundle of manifold, $T^*M$ is the cotangent bundle of the manifold.
    $$ TM=bigsqcup_{pin M} T_pM$$
    subject to the "gluing" conditions that $TM$ is "locally" of the form $Utimes mathbb{R}^n$ for $U$ an open neighborhood of $M$. These objects are called smooth vector bundles and you can read more about them in a book like



    $(1)$ Lee's Introduction to Smooth Manifolds



    $(2)$ Tu's An Introduction to Manifolds.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      $T_pM$ is the space of vectors tangent to $M$ at $p$, and $T_p^*M=(T_pM)^*$ is the corresponding dual space.



      $TM$ is the tangent bundle of manifold, $T^*M$ is the cotangent bundle of the manifold.
      $$ TM=bigsqcup_{pin M} T_pM$$
      subject to the "gluing" conditions that $TM$ is "locally" of the form $Utimes mathbb{R}^n$ for $U$ an open neighborhood of $M$. These objects are called smooth vector bundles and you can read more about them in a book like



      $(1)$ Lee's Introduction to Smooth Manifolds



      $(2)$ Tu's An Introduction to Manifolds.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        $T_pM$ is the space of vectors tangent to $M$ at $p$, and $T_p^*M=(T_pM)^*$ is the corresponding dual space.



        $TM$ is the tangent bundle of manifold, $T^*M$ is the cotangent bundle of the manifold.
        $$ TM=bigsqcup_{pin M} T_pM$$
        subject to the "gluing" conditions that $TM$ is "locally" of the form $Utimes mathbb{R}^n$ for $U$ an open neighborhood of $M$. These objects are called smooth vector bundles and you can read more about them in a book like



        $(1)$ Lee's Introduction to Smooth Manifolds



        $(2)$ Tu's An Introduction to Manifolds.






        share|cite|improve this answer









        $endgroup$



        $T_pM$ is the space of vectors tangent to $M$ at $p$, and $T_p^*M=(T_pM)^*$ is the corresponding dual space.



        $TM$ is the tangent bundle of manifold, $T^*M$ is the cotangent bundle of the manifold.
        $$ TM=bigsqcup_{pin M} T_pM$$
        subject to the "gluing" conditions that $TM$ is "locally" of the form $Utimes mathbb{R}^n$ for $U$ an open neighborhood of $M$. These objects are called smooth vector bundles and you can read more about them in a book like



        $(1)$ Lee's Introduction to Smooth Manifolds



        $(2)$ Tu's An Introduction to Manifolds.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 6 at 22:26









        Antonios-Alexandros RobotisAntonios-Alexandros Robotis

        10.3k41641




        10.3k41641






























            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%2f3064443%2fwhat-does-tm-tm-and-meandefinition-check%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

            Human spaceflight

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            張江高科駅