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






differential-geometry notation definition







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$endgroup$












  • $begingroup$
    $TM$ 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






differential-geometry notation definition







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






differential-geometry notation definition







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




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






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oldest

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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









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    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

        9,75241640




        9,75241640






























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