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1. • CommentRowNumber1.
   • CommentAuthorfred53
   • CommentTimeMar 17th 2019
   • (edited Mar 17th 2019)
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   Author: fred53
   Format: TextI have written a 20 page expository article on differential geometry at an advanced undergraduate level. I wonder if I could post the article on nLab. Here, I copy from the introduction of the paper, and would appreciate any comments. This twenty page note aims at a clear and quick exposition of some basic concepts and results in differential geometry, starting from the definition of vector fields, and culminating in Hodge theory on Kahler manifolds. Any success comes at the expense of omitting all proofs as well as key tools like sheaf theory (except in passing remarks) and pull back functions and their functorial properties. I have tried and believe to have make the prerequisites few and the exposition simple. Researching for this note helped me consolidate foggy recollections of my decades-old studies, and I hope it will likewise prove useful to some readers in their learning introductory differential geometry. I assume the reader knows how real and complex manifolds and occasionally vector bundles are defined, but beyond this the development is self contained. It concentrates on the algebra $\A$ (or $\A_\C$) of smooth real (or complex) valued functions on the manifold, viewing tensors, forms and indeed smooth sections of all vector bundles as $\A$ (or $\A_\C$) modules. Nothing in commutative algebra harder than the concept of module homomorphism (which I call $\A$-linear) and its multilinear counterpart is used, yet this simple language goes a long way to economize our presentation. The pace is leisurely in the beginning for the benefit of the novice, then picks up a bit in later sections. The first 6 sections are about real smooth manifolds, sections 7 and 8 discuss real and complex vector bundles over real manifolds, and the final 3 sections are about complex manifolds. I start by first defining vector fields, tensor fields, Lie derivative and then move on to metrics and (Levi-Civita) connections on the tangent bundle and their Riemann, Ricci and scalar curvature. Sec 5 defines differential forms and lists their main properties. Sec 6 discusses Hodge theory and harmonic forms on real manifolds. Sec 7 is about connections and their curvature on real vector bundles and Bianchi identities and Sec 8 presents complex vector bundles on real manifolds and their Chern classes. Sec 9 discusses complex manifolds and the Dolbeault complex and Sec 10 Chern connections on holoromorphic vector bundles. Sec 11 discusses the Hodge decomposition on compact Kahler manifolds. Beyond whatever left of my college day studies, I have drawn freely from internet sources, including nLab, particularly Wikepedia, as well as some downloadable books and notes. I give no references because aside from my own expository peculiarities, choices, typos, or any errors, the material is textbook standard.

   I have written a 20 page expository article on differential geometry at an advanced undergraduate level. I wonder if I could post the article on nLab. Here, I copy from the introduction of the paper, and would appreciate any comments.
   
   This twenty page note aims at a clear and quick exposition of some basic concepts and results in differential geometry, starting from the definition of vector fields, and culminating in Hodge theory on Kahler manifolds. Any success comes at the expense of omitting all proofs as well as key tools like sheaf theory (except in passing remarks) and pull back functions and their functorial properties. I have tried and believe to have make the prerequisites few and the exposition simple. Researching for this note helped me consolidate foggy recollections of my decades-old studies, and
   I hope it will likewise prove useful to some readers in their learning introductory differential geometry.
   
   I assume the reader knows how real and complex manifolds and occasionally vector bundles are defined, but beyond this the development is self contained. It concentrates on the algebra $\A$ (or $\A_\C$) of smooth real (or complex) valued functions on the manifold, viewing tensors, forms and indeed smooth sections of all vector bundles as $\A$ (or $\A_\C$) modules. Nothing in commutative algebra harder than the concept of module homomorphism (which I call $\A$-linear) and its multilinear counterpart is used, yet this simple language goes a long way to economize our presentation.
   The pace is leisurely in the beginning for the benefit of the novice, then picks up a bit in later sections.
   
   The first 6 sections are about real smooth manifolds, sections 7 and 8 discuss real and complex vector bundles over real manifolds, and the final 3 sections are about complex manifolds. I start by first defining vector fields, tensor fields, Lie derivative and then move on to metrics and (Levi-Civita) connections on the tangent bundle and their Riemann, Ricci and scalar curvature. Sec 5 defines differential forms and lists their main properties. Sec 6 discusses Hodge theory and harmonic forms on real manifolds. Sec 7 is about connections and their curvature on real vector bundles and Bianchi identities and Sec 8 presents complex vector bundles on real manifolds and their Chern classes. Sec 9 discusses complex manifolds and the Dolbeault complex and Sec 10 Chern connections on holoromorphic vector bundles. Sec 11 discusses the Hodge decomposition on compact Kahler manifolds.
   
   Beyond whatever left of my college day studies, I have drawn freely from internet sources, including nLab, particularly Wikepedia, as well as some downloadable books and notes. I give no references because aside from my own expository peculiarities, choices, typos, or any errors, the material is textbook standard.
2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 17th 2019
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   Author: DavidRoberts
   Format: MarkdownItexAre you able to put in the cloud somewhere and link to it here? It's hard to know imagine what it looks like, 20 pages is not much to go from novice to the Hodge decomposition.

   Are you able to put in the cloud somewhere and link to it here? It’s hard to know imagine what it looks like, 20 pages is not much to go from novice to the Hodge decomposition.

3. • CommentRowNumber3.
   • CommentAuthorfred53
   • CommentTimeMar 17th 2019
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   Author: fred53
   Format: TextI put it on research gate. Can you access it? https://www.researchgate.net/publication/331813344_fj_notes_on_differential_geometry

   I put it on research gate. Can you access it? https://www.researchgate.net/publication/331813344_fj_notes_on_differential_geometry
4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 17th 2019
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   Author: DavidRoberts
   Format: MarkdownItexYes, it's visible. Now people can have a look and give an informed responce to your question.

   Yes, it’s visible. Now people can have a look and give an informed responce to your question.

5. • CommentRowNumber5.
   • CommentAuthorfred53
   • CommentTimeMar 17th 2019
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   Author: fred53
   Format: TextThanks, I updated Sec 5. I hope some people will look at the link and comment on its suitability. Meanwhile I'll try to learn how enter it as an nLab article with a link, if nobody objects.

   Thanks, I updated Sec 5. I hope some people will look at the link and comment on its suitability. Meanwhile I'll try to learn how enter it as an nLab article with a link, if nobody objects.
6. • CommentRowNumber6.
   • CommentAuthorfred53
   • CommentTimeMar 17th 2019
   • PermaLink
   Author: fred53
   Format: TextWell, I created an nLab page and put a link to my expository article. See nLab page https://ncatlab.org/nlab/show/Expository%20article%20on%20differential%20geometry

   Well, I created an nLab page and put a link to my expository article. See nLab page https://ncatlab.org/nlab/show/Expository%20article%20on%20differential%20geometry
7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 17th 2019
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   Author: DavidRoberts
   Format: MarkdownItexSee my [comments on the other thread](https://nforum.ncatlab.org/discussion/9690/expository-article-on-differential-geometry/#Item_4).

   See my comments on the other thread.

8. • CommentRowNumber8.
   • CommentAuthorfred53
   • CommentTimeMar 17th 2019
   • PermaLink
   Author: fred53
   Format: TextI created an nLab page and put a link to my expository article. See nLab page [[Expository article on differential geometry]]

   I created an nLab page and put a link to my expository article.
   See nLab page Expository article on differential geometry
9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMar 18th 2019
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   Author: Urs
   Format: MarkdownItexHi Farshid, I was busy last week. Just looked at you article [here](https://www.researchgate.net/publication/331813344_fj_notes_on_differential_geometry). As somebody else noted, it seems to be not so much an exposition than a fairly bare list of statements of some basic definitions. I would say in its present form this does not justify a standalone nLab page. Instead, you should just edit the corresponding separate pages where you see the need! For instance the explicit formula for the Lie derivative which your note recalls is curently not stated in the entry _[[Lie derivative]]_. A useful edit would be to add that definition there in that entry. In editing entries, please try to interact a little with the nature of the $n$Lab. For instance, it is funny to have an $n$Lab entry start out by saying that it's 20 pages long. Because, first of all this is hardly the relevant information the entry wants to convey, second it makes no good sense on a wiki, and even if it did, it would change with time the moment anyone else adds to the entry. In conclusion, I suggest that for the time being you step back from the idea of writing an "Expository article" on the $n$Lab and instead first add basic definitions to relevant existing $n$Lab entries, where need be. Such editing will eventually give you a feeling for the nature of the $n$Lab, and when that has happened it may be time to come back to the idea of creating an expository entry of DG.

   Hi Farshid,

   I was busy last week. Just looked at you article here.

   As somebody else noted, it seems to be not so much an exposition than a fairly bare list of statements of some basic definitions. I would say in its present form this does not justify a standalone nLab page. Instead, you should just edit the corresponding separate pages where you see the need!

   For instance the explicit formula for the Lie derivative which your note recalls is curently not stated in the entry Lie derivative. A useful edit would be to add that definition there in that entry.

   In editing entries, please try to interact a little with the nature of the $n$Lab. For instance, it is funny to have an $n$Lab entry start out by saying that it’s 20 pages long. Because, first of all this is hardly the relevant information the entry wants to convey, second it makes no good sense on a wiki, and even if it did, it would change with time the moment anyone else adds to the entry.

   In conclusion, I suggest that for the time being you step back from the idea of writing an “Expository article” on the $n$Lab and instead first add basic definitions to relevant existing $n$Lab entries, where need be.

   Such editing will eventually give you a feeling for the nature of the $n$Lab, and when that has happened it may be time to come back to the idea of creating an expository entry of DG.

10. • CommentRowNumber10.
    • CommentAuthorfred53
    • CommentTimeMar 18th 2019
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    Author: fred53
    Format: TextHi Urs, Thank you for your comments. Yes, I guess the paper is more a compilation of facts in a logical order than expository. It seems still instructive to see how far you can quickly go when you omit proofs and certain definitions. My aim was simplicity and readability while being (more or less) self contained. I realize that nLab, working at the cutting edge, demands more and would keep that in mind should I take on your suggestion of editing some nLab entries. My "introduction" that mentioned the twenty pages is not part of the paper proper. Rather, since nLab recommends an "Idea" section, I put an introduction there to explain its motivation, as well as in ResearchGate's Description, both outside the paper and not distributed with the pdf. Thanks also for all the good work that you and your colleagues do at nLab.

    Hi Urs,
    
    Thank you for your comments. Yes, I guess the paper is more a compilation of facts in a logical order than expository. It seems still instructive to see how far you can quickly go when you omit proofs and certain definitions. My aim was simplicity and readability while being (more or less) self contained. I realize that nLab, working at the cutting edge, demands more and would keep that in mind should I take on your suggestion of editing some nLab entries. My "introduction" that mentioned the twenty pages is not part of the paper proper. Rather, since nLab recommends an "Idea" section, I put an introduction there to explain its motivation, as well as in ResearchGate's Description, both outside the paper and not distributed with the pdf. Thanks also for all the good work that you and your colleagues do at nLab.
11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2019
    • (edited Mar 18th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> I realize that nLab, working at the cutting edge, demands more No, that wasn't what we said; basic stuff is more than welcome for the $n$Lab. For instance that explicit definition of the Lie bracket of vector fields fully deserves to be on the $n$Lab, as I said, and we'd appreciate it if you'd added it. What we complained about was the formatting, not the content; that you pasted a pdf text into the $n$Lab without interacting with the difference between a wiki and a private pdf. If you don't mean to create an exposition, but something like a "list of basic definitions in differential geometry", that might well deserve an $n$Lab entry of its own, after all. An appropriate title for such an entry might be "basic definitions in differential geometry". But before starting that, you could just begin to add such a list to the existing entry _[[differential geometry]]_ and see how that goes. When and if there is enough material added to justify splitting off as a standalone entry, you can still do it.

    I realize that nLab, working at the cutting edge, demands more

    No, that wasn’t what we said; basic stuff is more than welcome for the $n$Lab. For instance that explicit definition of the Lie bracket of vector fields fully deserves to be on the $n$Lab, as I said, and we’d appreciate it if you’d added it.

    What we complained about was the formatting, not the content; that you pasted a pdf text into the $n$Lab without interacting with the difference between a wiki and a private pdf.

    If you don’t mean to create an exposition, but something like a “list of basic definitions in differential geometry”, that might well deserve an $n$Lab entry of its own, after all. An appropriate title for such an entry might be “basic definitions in differential geometry”. But before starting that, you could just begin to add such a list to the existing entry differential geometry and see how that goes. When and if there is enough material added to justify splitting off as a standalone entry, you can still do it.

12. • CommentRowNumber12.
    • CommentAuthorfred53
    • CommentTimeMar 18th 2019
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    Author: fred53
    Format: TextThanks for your further comments. I may well pick up on your suggestion later. But, at present, I wouldn't just want to add a list of definitions - I'd also want to add properties and results, as in the paper. So far, my main aim has been to expose the novice to some of the main concepts and results in differential geometry in a quick and readable way.

    Thanks for your further comments. I may well pick up on your suggestion later. But, at present, I wouldn't just want to add a list of definitions - I'd also want to add properties and results, as in the paper. So far, my main aim has been to expose the novice to some of the main concepts and results in differential geometry in a quick and readable way.