Mixing
Literature on Chern-Weil Theory and the Chern-Gauß-Bonnet Theorem
$begingroup$
At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and modern literature on this subject. Classically I know of Milnor's Appendix C in Milnor, Stasheff: "Characteristic Classes" and the famous text from Spivak's A Comprehensive Introduction to Differential Geometry entitled "The Generalized Gauß-Bonnet Theorem and What It Means To Mankind". Is there any more recent recommendable source?
During my search I also came along Zhang: Lectures on Chern-Weil Theory and Witten Deformations. Does anyone have practical experience with this book?
Thanks in advance!
differential-geometry soft-question advice
asked Feb 10 '14 at 22:56
user52942user52942
435
$endgroup$
add a comment |
$begingroup$
At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and modern literature on this subject. Classically I know of Milnor's Appendix C in Milnor, Stasheff: "Characteristic Classes" and the famous text from Spivak's A Comprehensive Introduction to Differential Geometry entitled "The Generalized Gauß-Bonnet Theorem and What It Means To Mankind". Is there any more recent recommendable source?
During my search I also came along Zhang: Lectures on Chern-Weil Theory and Witten Deformations. Does anyone have practical experience with this book?
Thanks in advance!
differential-geometry soft-question advice
asked Feb 10 '14 at 22:56
user52942user52942
435
$endgroup$
2
$begingroup$
Zhang's lecture notes will be useful to get aware of everything, but it's pretty hand-wavy without much details. If that's all you need, then Zhang's fine.
$endgroup$
– Chris Gerig
Feb 10 '14 at 23:04
$begingroup$
This material appears in many standard textbooks (Zhang's notes is not a standard reference). You can also look at the original articles of Chern. There is an article by Chern and Bott on equidistribution of zeros of sections of holomorphic vector bundles that begins with a nice summary of the Chern-Weil theory. The Wikipedia article on the Chern–Weil homomorphism has a fine reference list.
$endgroup$
– Gil Bor
Feb 12 '14 at 15:01
$begingroup$
Volume II of "Connections, Curvature and Cohomology" by Greub et al. covers the Gauss-Bonnet-Chern theorem and the Chern-Weil homomorphism.
$endgroup$
– archipelago
Feb 12 '14 at 17:49
$begingroup$
There are a bunch of commented references at the end of ncatlab.org/nlab/show/Chern-Weil+theory
$endgroup$
– archipelago
Feb 12 '14 at 19:33
add a comment |
$begingroup$
At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and modern literature on this subject. Classically I know of Milnor's Appendix C in Milnor, Stasheff: "Characteristic Classes" and the famous text from Spivak's A Comprehensive Introduction to Differential Geometry entitled "The Generalized Gauß-Bonnet Theorem and What It Means To Mankind". Is there any more recent recommendable source?
During my search I also came along Zhang: Lectures on Chern-Weil Theory and Witten Deformations. Does anyone have practical experience with this book?
Thanks in advance!
differential-geometry soft-question advice
asked Feb 10 '14 at 22:56
user52942user52942
435
$endgroup$
At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and modern literature on this subject. Classically I know of Milnor's Appendix C in Milnor, Stasheff: "Characteristic Classes" and the famous text from Spivak's A Comprehensive Introduction to Differential Geometry entitled "The Generalized Gauß-Bonnet Theorem and What It Means To Mankind". Is there any more recent recommendable source?
During my search I also came along Zhang: Lectures on Chern-Weil Theory and Witten Deformations. Does anyone have practical experience with this book?
Thanks in advance!
differential-geometry soft-question advice
differential-geometry soft-question advice
asked Feb 10 '14 at 22:56
user52942user52942
435
asked Feb 10 '14 at 22:56
user52942user52942
435
asked Feb 10 '14 at 22:56
user52942user52942
435
asked Feb 10 '14 at 22:56
user52942user52942
435
asked Feb 10 '14 at 22:56
user52942user52942
435
435
2
$begingroup$
Zhang's lecture notes will be useful to get aware of everything, but it's pretty hand-wavy without much details. If that's all you need, then Zhang's fine.
$endgroup$
– Chris Gerig
Feb 10 '14 at 23:04
$begingroup$
This material appears in many standard textbooks (Zhang's notes is not a standard reference). You can also look at the original articles of Chern. There is an article by Chern and Bott on equidistribution of zeros of sections of holomorphic vector bundles that begins with a nice summary of the Chern-Weil theory. The Wikipedia article on the Chern–Weil homomorphism has a fine reference list.
$endgroup$
– Gil Bor
Feb 12 '14 at 15:01
$begingroup$
Volume II of "Connections, Curvature and Cohomology" by Greub et al. covers the Gauss-Bonnet-Chern theorem and the Chern-Weil homomorphism.
$endgroup$
– archipelago
Feb 12 '14 at 17:49
$begingroup$
There are a bunch of commented references at the end of ncatlab.org/nlab/show/Chern-Weil+theory
$endgroup$
– archipelago
Feb 12 '14 at 19:33
add a comment |
2
$begingroup$
Zhang's lecture notes will be useful to get aware of everything, but it's pretty hand-wavy without much details. If that's all you need, then Zhang's fine.
$endgroup$
– Chris Gerig
Feb 10 '14 at 23:04
$begingroup$
This material appears in many standard textbooks (Zhang's notes is not a standard reference). You can also look at the original articles of Chern. There is an article by Chern and Bott on equidistribution of zeros of sections of holomorphic vector bundles that begins with a nice summary of the Chern-Weil theory. The Wikipedia article on the Chern–Weil homomorphism has a fine reference list.
$endgroup$
– Gil Bor
Feb 12 '14 at 15:01
$begingroup$
Volume II of "Connections, Curvature and Cohomology" by Greub et al. covers the Gauss-Bonnet-Chern theorem and the Chern-Weil homomorphism.
$endgroup$
– archipelago
Feb 12 '14 at 17:49
$begingroup$
There are a bunch of commented references at the end of ncatlab.org/nlab/show/Chern-Weil+theory
$endgroup$
– archipelago
Feb 12 '14 at 19:33
2
2
$begingroup$
Zhang's lecture notes will be useful to get aware of everything, but it's pretty hand-wavy without much details. If that's all you need, then Zhang's fine.
$endgroup$
– Chris Gerig
Feb 10 '14 at 23:04
$begingroup$
Zhang's lecture notes will be useful to get aware of everything, but it's pretty hand-wavy without much details. If that's all you need, then Zhang's fine.
$endgroup$
– Chris Gerig
Feb 10 '14 at 23:04
$begingroup$
This material appears in many standard textbooks (Zhang's notes is not a standard reference). You can also look at the original articles of Chern. There is an article by Chern and Bott on equidistribution of zeros of sections of holomorphic vector bundles that begins with a nice summary of the Chern-Weil theory. The Wikipedia article on the Chern–Weil homomorphism has a fine reference list.
$endgroup$
– Gil Bor
Feb 12 '14 at 15:01
$begingroup$
This material appears in many standard textbooks (Zhang's notes is not a standard reference). You can also look at the original articles of Chern. There is an article by Chern and Bott on equidistribution of zeros of sections of holomorphic vector bundles that begins with a nice summary of the Chern-Weil theory. The Wikipedia article on the Chern–Weil homomorphism has a fine reference list.
$endgroup$
– Gil Bor
Feb 12 '14 at 15:01
$begingroup$
Volume II of "Connections, Curvature and Cohomology" by Greub et al. covers the Gauss-Bonnet-Chern theorem and the Chern-Weil homomorphism.
$endgroup$
– archipelago
Feb 12 '14 at 17:49
$begingroup$
Volume II of "Connections, Curvature and Cohomology" by Greub et al. covers the Gauss-Bonnet-Chern theorem and the Chern-Weil homomorphism.
$endgroup$
– archipelago
Feb 12 '14 at 17:49
$begingroup$
There are a bunch of commented references at the end of ncatlab.org/nlab/show/Chern-Weil+theory
$endgroup$
– archipelago
Feb 12 '14 at 19:33
$begingroup$
There are a bunch of commented references at the end of ncatlab.org/nlab/show/Chern-Weil+theory
$endgroup$
– archipelago
Feb 12 '14 at 19:33
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
For reference's sake, here is Chern's original paper:
http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf
The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other papers related to Chern class and Gauss-Bonnet theorem, which you can look up.
One should note that the notations in Chern's paper is slightly different from the ones used in contemporary Riemannian geometry textbooks. You need to check Chern's Lectures on Differential geometry as a reference .
For the most trivial example, he define Christoffel symbols by
$$
nabla s_{alpha}=Gamma^{beta}_{alpha i}du_{i}otimes s_{beta}, omega^{beta}_{alpha}=Gamma^{beta}_{alpha i}du_{i}
$$
so we have
$$
nabla s_{alpha}=w^{beta}_{alpha}otimes s_{beta}
$$
and if one use $S=[s_{1}cdots s_{n}]^{T}$, $omega={omega_{ij}}$, then we can abbreviate this as
$$
nabla S=omegaotimes S
$$
and it only takes one line from his paper because he assumed that reader knows about it. However, I do not think this is explained very clearly in "modern style" textbooks (Jost, Lee, Taubes, etc). And this might make the reader reading his paper difficult.
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
$endgroup$
add a comment |
$begingroup$
This book was not available four years ago but I thought I should include this here for posterity. There's this book by Loring W. Tu published recently in 2017 titled Differential Geometry: Connections, Curvature, and Characteristic Classes. It provides a really accessible introduction to Chern-Weil Theory, though the Chern-Gauß-Bonnet Theorem was only stated without proof.
answered Jan 10 at 4:29
A. H.A. H.
11
$endgroup$
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f671543%2fliterature-on-chern-weil-theory-and-the-chern-gau%25c3%259f-bonnet-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For reference's sake, here is Chern's original paper:
http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf
The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other papers related to Chern class and Gauss-Bonnet theorem, which you can look up.
One should note that the notations in Chern's paper is slightly different from the ones used in contemporary Riemannian geometry textbooks. You need to check Chern's Lectures on Differential geometry as a reference .
For the most trivial example, he define Christoffel symbols by
$$
nabla s_{alpha}=Gamma^{beta}_{alpha i}du_{i}otimes s_{beta}, omega^{beta}_{alpha}=Gamma^{beta}_{alpha i}du_{i}
$$
so we have
$$
nabla s_{alpha}=w^{beta}_{alpha}otimes s_{beta}
$$
and if one use $S=[s_{1}cdots s_{n}]^{T}$, $omega={omega_{ij}}$, then we can abbreviate this as
$$
nabla S=omegaotimes S
$$
and it only takes one line from his paper because he assumed that reader knows about it. However, I do not think this is explained very clearly in "modern style" textbooks (Jost, Lee, Taubes, etc). And this might make the reader reading his paper difficult.
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
$endgroup$
add a comment |
$begingroup$
For reference's sake, here is Chern's original paper:
http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf
The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other papers related to Chern class and Gauss-Bonnet theorem, which you can look up.
One should note that the notations in Chern's paper is slightly different from the ones used in contemporary Riemannian geometry textbooks. You need to check Chern's Lectures on Differential geometry as a reference .
For the most trivial example, he define Christoffel symbols by
$$
nabla s_{alpha}=Gamma^{beta}_{alpha i}du_{i}otimes s_{beta}, omega^{beta}_{alpha}=Gamma^{beta}_{alpha i}du_{i}
$$
so we have
$$
nabla s_{alpha}=w^{beta}_{alpha}otimes s_{beta}
$$
and if one use $S=[s_{1}cdots s_{n}]^{T}$, $omega={omega_{ij}}$, then we can abbreviate this as
$$
nabla S=omegaotimes S
$$
and it only takes one line from his paper because he assumed that reader knows about it. However, I do not think this is explained very clearly in "modern style" textbooks (Jost, Lee, Taubes, etc). And this might make the reader reading his paper difficult.
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
$endgroup$
add a comment |
$begingroup$
For reference's sake, here is Chern's original paper:
http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf
The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other papers related to Chern class and Gauss-Bonnet theorem, which you can look up.
One should note that the notations in Chern's paper is slightly different from the ones used in contemporary Riemannian geometry textbooks. You need to check Chern's Lectures on Differential geometry as a reference .
For the most trivial example, he define Christoffel symbols by
$$
nabla s_{alpha}=Gamma^{beta}_{alpha i}du_{i}otimes s_{beta}, omega^{beta}_{alpha}=Gamma^{beta}_{alpha i}du_{i}
$$
so we have
$$
nabla s_{alpha}=w^{beta}_{alpha}otimes s_{beta}
$$
and if one use $S=[s_{1}cdots s_{n}]^{T}$, $omega={omega_{ij}}$, then we can abbreviate this as
$$
nabla S=omegaotimes S
$$
and it only takes one line from his paper because he assumed that reader knows about it. However, I do not think this is explained very clearly in "modern style" textbooks (Jost, Lee, Taubes, etc). And this might make the reader reading his paper difficult.
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
$endgroup$
For reference's sake, here is Chern's original paper:
http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf
The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other papers related to Chern class and Gauss-Bonnet theorem, which you can look up.
One should note that the notations in Chern's paper is slightly different from the ones used in contemporary Riemannian geometry textbooks. You need to check Chern's Lectures on Differential geometry as a reference .
For the most trivial example, he define Christoffel symbols by
$$
nabla s_{alpha}=Gamma^{beta}_{alpha i}du_{i}otimes s_{beta}, omega^{beta}_{alpha}=Gamma^{beta}_{alpha i}du_{i}
$$
so we have
$$
nabla s_{alpha}=w^{beta}_{alpha}otimes s_{beta}
$$
and if one use $S=[s_{1}cdots s_{n}]^{T}$, $omega={omega_{ij}}$, then we can abbreviate this as
$$
nabla S=omegaotimes S
$$
and it only takes one line from his paper because he assumed that reader knows about it. However, I do not think this is explained very clearly in "modern style" textbooks (Jost, Lee, Taubes, etc). And this might make the reader reading his paper difficult.
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
answered Aug 4 '14 at 21:41
Bombyx moriBombyx mori
13k63075
13k63075
add a comment |
add a comment |
$begingroup$
This book was not available four years ago but I thought I should include this here for posterity. There's this book by Loring W. Tu published recently in 2017 titled Differential Geometry: Connections, Curvature, and Characteristic Classes. It provides a really accessible introduction to Chern-Weil Theory, though the Chern-Gauß-Bonnet Theorem was only stated without proof.
answered Jan 10 at 4:29
A. H.A. H.
11
$endgroup$
add a comment |
$begingroup$
This book was not available four years ago but I thought I should include this here for posterity. There's this book by Loring W. Tu published recently in 2017 titled Differential Geometry: Connections, Curvature, and Characteristic Classes. It provides a really accessible introduction to Chern-Weil Theory, though the Chern-Gauß-Bonnet Theorem was only stated without proof.
answered Jan 10 at 4:29
A. H.A. H.
11
$endgroup$
add a comment |
$begingroup$
This book was not available four years ago but I thought I should include this here for posterity. There's this book by Loring W. Tu published recently in 2017 titled Differential Geometry: Connections, Curvature, and Characteristic Classes. It provides a really accessible introduction to Chern-Weil Theory, though the Chern-Gauß-Bonnet Theorem was only stated without proof.
answered Jan 10 at 4:29
A. H.A. H.
11
$endgroup$
This book was not available four years ago but I thought I should include this here for posterity. There's this book by Loring W. Tu published recently in 2017 titled Differential Geometry: Connections, Curvature, and Characteristic Classes. It provides a really accessible introduction to Chern-Weil Theory, though the Chern-Gauß-Bonnet Theorem was only stated without proof.
answered Jan 10 at 4:29
A. H.A. H.
11
answered Jan 10 at 4:29
A. H.A. H.
11
answered Jan 10 at 4:29
A. H.A. H.
11
answered Jan 10 at 4:29
A. H.A. H.
11
11
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f671543%2fliterature-on-chern-weil-theory-and-the-chern-gau%25c3%259f-bonnet-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
2
$begingroup$
Zhang's lecture notes will be useful to get aware of everything, but it's pretty hand-wavy without much details. If that's all you need, then Zhang's fine.
$endgroup$
– Chris Gerig
Feb 10 '14 at 23:04
$begingroup$
This material appears in many standard textbooks (Zhang's notes is not a standard reference). You can also look at the original articles of Chern. There is an article by Chern and Bott on equidistribution of zeros of sections of holomorphic vector bundles that begins with a nice summary of the Chern-Weil theory. The Wikipedia article on the Chern–Weil homomorphism has a fine reference list.
$endgroup$
– Gil Bor
Feb 12 '14 at 15:01
$begingroup$
Volume II of "Connections, Curvature and Cohomology" by Greub et al. covers the Gauss-Bonnet-Chern theorem and the Chern-Weil homomorphism.
$endgroup$
– archipelago
Feb 12 '14 at 17:49
$begingroup$
There are a bunch of commented references at the end of ncatlab.org/nlab/show/Chern-Weil+theory
$endgroup$
– archipelago
Feb 12 '14 at 19:33