What does $TM$, $T^*M$ and $*$ mean?(Definition Check)
$begingroup$
I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.
However, I encountered some notation confusion:
What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?
What's the difference between $TM$ and $T^*M$?
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:
Image II:
differential-geometry notation definition
asked Jan 6 at 22:15
user9976437user9976437
759
$endgroup$
|
show 1 more comment
$begingroup$
I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.
However, I encountered some notation confusion:
What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?
What's the difference between $TM$ and $T^*M$?
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:
Image II:
differential-geometry notation definition
asked Jan 6 at 22:15
user9976437user9976437
759
$endgroup$
$begingroup$
$TM$ is the tangent bundle of the manifold $M$.
$endgroup$
– Matt Samuel
Jan 6 at 22:18
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@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
|
show 1 more comment
$begingroup$
I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.
However, I encountered some notation confusion:
What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?
What's the difference between $TM$ and $T^*M$?
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:
Image II:
differential-geometry notation definition
asked Jan 6 at 22:15
user9976437user9976437
759
$endgroup$
I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.
However, I encountered some notation confusion:
What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?
What's the difference between $TM$ and $T^*M$?
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:
Image II:
differential-geometry notation definition
differential-geometry notation definition
asked Jan 6 at 22:15
user9976437user9976437
759
asked Jan 6 at 22:15
user9976437user9976437
759
asked Jan 6 at 22:15
user9976437user9976437
759
asked Jan 6 at 22:15
user9976437user9976437
759
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
|
show 1 more comment
$begingroup$
$TM$ 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
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
$T_pM$ 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.
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
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add a comment |
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$begingroup$
$T_pM$ 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.
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
$endgroup$
add a comment |
$begingroup$
$T_pM$ 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.
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
$endgroup$
add a comment |
$begingroup$
$T_pM$ 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.
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
$endgroup$
$T_pM$ 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.
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
answered Jan 6 at 22:26
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
10.3k41641
add a comment |
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$begingroup$
$TM$ 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