Thread 16769118

I hate it, it's too "dry", too "mathematical", he barely explains anything intuitively or didactically. He just throws definitions and theorems at you and calls it a day. It's the Rudin of Differential Geometry.

>>16769118 (OP)
>He just throws definitions and theorems at you and calls it a day
I hate books like this:
>proposition
>definition
>theorem
>proof
>lemma
>proof
>corollary
>proof
>more theorem
>more proof

no example, no exercise, no explanation, no application
Motherfuckers who write book in this format should be castrated

>>16769148
Worst of all is that those guys are being awarded and praised for writing like that. While a guy who writes a math textbook carefully explaining the theorems and definitions, giving examples and detailing the intuition behind a proof or concept is ostracized for writing a “non-rigorous” book.

>>16769148
>>16769160
Based
fuck examples
let's go right into what matters.

Just do the exercises at the end of the chapters to get familiar with the things.

>>16769210
Examples are pretty helpful for people without autism. This is especially the case when the basic theorems and lemmas are non-constructivist and they give you no real indication of a good way to actually apply the theorems to solve real problems.

>>16769118 (OP)
Thoughts on this then? https://m.youtube.com/watch?v=_mvjOoTieTk&pp=ygUjZGlmZmVyZW50aWFsIGdlb21ldHJ5IG5qIHdpbGRiZXJnZXI%3D

>>16769118 (OP)
This book is not even close to Rudin like.

The definition-theorem is how math books are structured, some level of explanation permitted, usually the hope is that the idea as stated alone is written clearly enough to get, ie trying to motivate stuff may feel like filler sometimes. Here I think woth stuff like first fundamental form or covariant derivative, is quite clear on the intuition.

I think this book is very concrete and makes use of a lot of examples and computation.

>>16769118 (OP)
rigorous books are always bad because the authors make small mistakes everywhere and they don't allow any context to notice the errors, because they try to be rigid and minimalistic but if you take it literally the texts dont make sense. Its like reading these heavily verbosed legal texts where every comma counts but the writer is larping and doesnt actually know how to write

>>16769596
Since there seems to be several people like this in the thread, it is a math book. The whole thing is doing proof, so it at least needs to show the proper way in doing so. If conveying the bare intuition was enough, there wouldn't be much point to these subjects. I'm not sure what you mean 'if you take it literally the text doesn't make sense', in regard to errors or in general?

OP, just pick a different book if you dislike it. Do carmos work has been reprinted by dover, that means it is cheaper than your mom

Why are you complaining?

>>16769906
>'if you take it literally the text doesn't make sense', in regard to errors or in general?
In regards to errors.
You can correct for small errors with context, which means repetition and examples so that helps you pick out errors from the author and ignore them.
Errors and also ambiguity of the language (human language) and also the use of shorthands or different terms for some object which have not been declared before.

>>16770209
Because the teacher is using his book for the DG course I'm currently doing. I just can't switch to another book.

>>16769210
>>16769517
>>16769906
>>16770209
I'll give you guys an example of what I'm talking about. Pic related shows on the left do Carmo's definition of an arc length, no examples, no geometric intuition being given, he just throws the definition and barely explains anything. On the right it's the same definition of an arc length by another book (Pressley's) where he begins by reminding you of what a length is in mathematics and providing the geometric intuition behind arc length before writing down the actual definition, and below he even gives examples but I didn't include them or the image would be too big, do Carmo doesn't give a single example and immediately just gives you the exercises of the chapter.

>>16770270
Do Camo's is a massive larper with an inferiority complex. He wants to be more rigorous, more minimalistic and do a more thorough extermination of anything that can be mistaken as examplanation.

>>16770255
Are you in Scandinavia?

>>16770275
Are you replying to yourself? I genuinely dont know anyone that'd have beef with do Carmo, like that reputation is for Lang.

>>16770382
>Are you replying to yourself?
no
>>16770382
>I genuinely dont know anyone that'd have beef with do Carmo,
Retarded macaco that thinks the harder a book is to understand the better it is. Some kind of macaco logic where bad=actually good.

>>16770394
ay some casual racism for a trite point.

The broader point is do Carmo is hardly on the level of 'harder = better' or 'rigor for rigor sake' book. It's not as simple a question of hard vs easy being better: easy can be bad for not really engageing you in the subject and making you think, hard can be bad for simply being impenetrable.

>>16770401
>is hardly on the level of 'harder = better' or 'rigor for rigor sake' book.
Its his whole act

>>16770270
This feels hardly representative, it's barely at the beginning of the book and barely in the substance of the subject. Calc 3 is a prereq so it's not unusual to expect arclength or the 'chop into small intervals' intuition to be well known and hence relegated to exercise. It also may be a difference in generation and the built expectations of students, usually new textbooks arise from history of teaching a class.

Scanning through though I'd say I actually feel do Carmo overall is more example driven and uses lots of diagrams for these examples, I mean there is more examples of parametrized curves given for instance, lots of different kinds of surfaces later on. Just scroll/flip through and see what I mean. Pressley seems to use diagrams ig for only the 'simple' examples before diving into lots of algebra it seems. I kind of feel most diagrams wouldn't be helpful.

I genuinely don't see this 'do Carmo too rigorous' or deal like nothing here strikes me as overly formalistic. Not that I am without my criticisms of do Carmo, but if anything I felt it acted in too classic and low level a fashion, lacking a nice tight theoretical framework.

>>16770407
do Carmo's book was literally written to portray things only requiring essentially linear algebra and calculus and it's very concrete. That hardly sounds to me like someone going for 'rigor for rigor sake'. I'm genuinely lost about this sentiment. All I can think is someone just doesn't like even the most basic of math rigor as an idea in general.

>>16769118 (OP)
“P.do”

>>16770409
Maybe I'm too stupid for differential geometry then, I just outright can't understand his book from section 1.5 onwards.

>>16770479
Have you tried O'Neill's Elementary Differential Geometry? I think it's a much better "first pass" at diff geo for people who are not really pure math people. They are also different enough in scope that you could do O'Neill first and then go back to do Carmo later for the topics that they cover differently.

>>16770255
I don't recall i was forced to use the books my professors recommended

>>16769118 (OP)
I too had certain "issues" with the book. Thing is, I don't recall too much specifically what it was. I think it was something like he didn't explain as much as he could IMO, and it happened fairly often.

Here in the picture provided is the only example I can think of; the proof itself is pretty simple. But then ONE PAGE LATER, he makes the remark that continuity is a must, when he never fucking mentions it in the proof at all. Took me a day or two to figure it out. Yes, it indeed true that continuity is required. So why is it, if you yourself haven't figured it out in 1-2 minute? Maybe I'm just dumb and it's really obvious to you guys. Like, for me, I think I was okay with the entire 1st chapter (it's only a 5 chapter book and the first 4 are the ones people should critique bare min, as the last is a lil diff). For this comment >>16770270, idk man, I personally don't love this as an example cause it feels simple enough to me. But maybe both of us aren't the target audience to this book, or maybe the book itself ain't amazing if many people are having issues, regardless of whether the particular examples of our difficulties are different.

If you haven't figured out why continuity is required, here is a "light" hint - light in that, to me, it isn't exactly a great hint that's help me out that much. But the issue INVOLVES (not exactly is) the hint. (don't read the greentext after this if you don't want the hint):

>INVERSE FUNCTION THEOREM
>Let [math] F: U \subset R^n \rightarrow R^n [/math] be a differentiable mapping and suppose that at [math] p \in U [/math] the differential [math] dF_p: R^n \rightarrow R^n [/math] is an isomorphism.
(means the matrix is invertible).
>Then there exists a neighborhood [math] V [/math] of [math] p [/math] in [math] U [/math] and a neighborhood [math] W [/math] of [math] F(p) [/math] in [math] R^n [/math] such that [math] F: V \rightarrow W [/math] has a differentiable inverse [math] F^{-1}: W \rightarrow V [/math]

>>16770518
Its because x and y are parametrizations, a concept explicitly defined in the last section which is stated to be a homeomorphism (hence continuous), or are differentiable which implies continuous.

It could be said that 'homeomorphisms' arent concepts introduced in calc: at beginning of section 2, says these concepts can be seen in a certain book on Advanced Calc and are given in the Appendix.

That to me is maybe the big difference: it hinges maybe on advanced calculus as being well known. Perhaps, back then such topics were commonly taught. My profs seem to think so. Maybe it is a question of more elite uni: in any case, I feel that's the missing context here-Advanced Calculus concepts from past courses due perhaps to poor teaching in Calculus 3. Just check out the Advanced Calculus book they cite in section 2 just to get an idea.

>>16770536
Where in the proof is it utilized that x and y inverses must be continuous? Where does it break to proof if not true? Nowhere does he mention why it's so important, but he does highly at least twice that it is very important. It is BECAUSE of this importance that, as you said, is
>a concept that is explicitly defined...

(I swear, im not spamming the page to update, I literally updated 57 seconds after your post by coincidence)

>>16770479
I don't think that way. Just identify these concepts.

I think it is a great idea to find external sources to supplement your understanding and something I wish I did more of rather than inherently forcing yourself through a text. However I don't think the critique of a book being too hard or rigorous is all that fair and I kind of feel it's a good idea to try to learn these concepts they use that makes it appear hard rather than avoid them, as these concepts tend to be helpful in the future. There is value to 'hard books'-you learn math by doing problems, especially hard ones. Hard books encourage this. Rudin does this but I also think it is quite fair, and some of its presentation makes the subject quite lucid, in particular the way I feel it encourages 'hard analysis' (not hard and soft as in difficulty but as concept: soft is more topological, hard is more analytic). However, would I expect it as a first pass in the subject? No, and something like Understanding Analysis by Abbott has value in helping centralize more the concepts. Plus other POVs: I really like its 'sequence first' POV on many concepts in analysis, for instance the notion of compactness of a set C as stating every sequence in C contains a limit in C, is just very pragmatically useful.

On the topic of including rigor, I can understand why some may not like it but I also feel it's fair to learn the subject 'how someone in that field would'. Like a math book is how a math person would, a physics book how a physics person would, an engineering book how an engineering person would, etc. To me it feels more authentic and humble. Rigor is a key part of math. I mean there would be no point to real analysis as a topic if we didn't care about such things. There are many 'quite natural functions' that break some rule you expect so it's natural to think about how to define such things.

>>16770540
like, there exists an importance for inverse continuity that is not explained. It is because of this importance that inverse continuity is part of the definitions. I already know the definitions of homeomorphism and parametrizations, but can you determine why or what the importance is?

Inverse continuity is "somehow" a requirement makes the proof true (can you figure it out?). The proof itself is very important for working with euclidean regular surfaces, and the statement itself (not proof) is very important when working with general manifolds. Thus, it is necessary to put inverse continuity into the definitions that were made in the beginning.

Hopefully this was explained better

>>16770540
NTA, but continuity is a good prerequisite in these arguments because (and this is a good definition of that it means to be a continuous function) a continuous functions f:X->Y has the property that the preimage of open sets in Y are open sets in X.


Without these, you really lose control as soon as you talk about the preimage of W.

>>16770548
yeah, I know, but the proof of that particular statement itself is the main thing the author identifies as important. Not saying it's the only perk, but it's a vital one, so much so that it affects the definitions of manifolds

>>16770547
Hmm, this is a good point.

For definitions like this I imagine it more like we have a list of properties we want this object to have based on intuitive conception, like how we want manifolds to look locally the 'same' as some section of R^n, topologically the idea of which is a homeomorphism. Its how we can guarantee transition maps are at least 'continuous'. Asking the real 'importance' of a property is good but sometimes, not all properties are spelled out in their use (see Proposition 4 below)

It might not be as 'spelled out' here: but differentiability is only defined on open sets. So, as W is open, and our function is defined on y^-1(W), we can only talk about differentiability of x^-1y if y^-1(W) is open. This is where x and y inverse being continuous is needed.

In any case, for the concepts used, Proposition 4 kind of highlights how x and y inverse being continuous is a 'superfluous' condition which is another reason I don't think it was explicitly used. Not saying this doesn't make your criticism valid but the proof at least is correct.

There's some deeper ideas going on here: first, abstract manifolds simply utilize charts as homeomorphisms. It's all topological. They are homeomorphisms so to as to guarantee the transition map is a homeomorphism. As this defined on an abstract topological space M, we have no 'R^n' to talk about differentiability with x and y alone. Next, to define a smooth structure, require x^-1y to be smooth for all parametrizations x and y. So, this is why the definition is given as so based on x and y being homeomorphisms even if it falls out from the definition he uses in this specific instance. (think of his definition as the pragmatic case for surfaces)

Its easy to create some intuition for these concepts by giving examples that are limited to 3 dimensions, then go to general cases. This doesnt work always, some theorems may explicitly work only in higher dimensions. But still.
Then you introduce general concepts. You can also offer practical examples at higher dimensions using numerical examples although you can do visual representations.
At least its what i would want in an introductory text. Books for professionals are different, students cant be expected to be good at the thing they are studying.

>>16770572
>we can only talk about differentiability of x^-1y [sic] if y^-1(w) is open
From the OP pic
>It is not possible to conclude, by an analogous argument... [since] we do not yet know what it is meant by a differentiable function on S.

He mentions earlier that this statement he's proving is needed to define said differentiability on surfaces.

>>16770572
>superfluous
Uh, I myself wouldn't call it that. Prop 4, in the example he uses afterwards, is simply used as a verification tool for the function x. I see prop 4 that you posted is only a verification tool, not a statement that makes the inverse cont requirement superfluous.

Yes, I know the proof is correct, and it 100% requires inverse cont. else it does not work, and I know now why - but it certainly wasn't with the author's help. The author made it much more clear than I have (i mean, he had a whole book to explain everything) that this condition is vital, but the fact that he leaves out a single paragraph explaining where it pops up in the proof, and only mentions it an entire page after... like, wtf man?

I'm pretty sure this was the most egregious thing he did in the book because it's the only one I wrote down in my notes, but issues like this to lesser degree was sprinkled throughout the book, I think. I'm pretty sure everything can be figured out after a day or two, but did he really have to fucking do it for the one the he very blatantly expressed was very important? I think he also said something early on like, the study of diff geom was held back a couple years because people didn't recognize the importance of that statement (but cites nothing?). Then why not explain it thoroughly????

He could've given a simple paragraph in the proof to mention it. An even better author would've also given a simple counter example showing how the proof breaks without that condition. But nooooo, let's just omit it and mention a page later...

>>16770603
Apologies, I was mistaking the continuity of y and y^-1 (Doh)

The condition is that we need W to be open. It will be when x(U) and y(V) are, which happens when x and y are homeomorphisms. Then with y continuous, W is open so y^-1(W) is open so W is open and differentiability of h makes sense.

Beyond that, I don't actually see anywhere the continuity of y^-1 is used, however to me this isn't an issue as it is 'baked' into what happens when you require x and y to be differentiable, the far more important fact for this construction to work. I don't really see how you can highlight where x or y being homeomorphisms comes up if it essentially doesn't.

To convince yourself of this difficulty, consider more abstract manifolds: where do 'homeomorphisms' come up in showing x^-1 y is differentiable? Trick question: you simply assume x^-1 y it is and this condition itself doesn't inherently need x and y to be homeomorphisms (find an example). Instead, its the other conditions where you utilize the fact x and y are homeomorphisms elsewhere in the rest of the theory of manifolds: but the concept of differentiability of transition maps doesn't in itself *require* it. It's an extra condition added on top the theory of manifolds.

Same anon as here>>16770548


>>16770626

If you're convinced now that we used that y is continuous because we needed that y^-1 sends open sets to open sets, then note that the last paragraph of the proof says we can rewrite the whole argument but now for h^-1 instead of h.

for *that* argument you'd need y^-1 to be continuous

>>16770612
Can you establish exactly where it *does* require inverse continuity and what is mentioned a page later?

I agree calling it superfluous is wrong but my thinking here is what you require to make the theory of surfaces tick. Differentiability of x, y is of utmost importance, to give a pragmatic means in theory of surfaces of constructing differentiable transition maps x^-1 y, something you normally have to 'postulate' in theory of smooth manifolds. It is usually a little more difficult to establish inverse continuity because surfaces as a subspace of R^3 are subject to constraints although in simple cases can solve for the inverse and show it is continuous. This question of 'pragmatic definition' is to me that comes up in another example: like in most theoretical definitions of a manifold M, it is assumed to have a topology already. However, charts themselves on a set M can generate a particular topology, meaning writing a chart is essentially good enough.

Also it would be impossible to give a counterexample because as stated, differentiability of x and y *necessitate* x and y to be homeomorphisms.

>>16770631
That isn't true.

h=xy^-1, so h^-1=yx^-1. This is just swapping x and y, so if none of x or y inverse were used in the proof of differentiability of h, it won't be used in h^-1. As it is now x^-1 on the right, we need continuity of x, not y.

>>16770638
>we need continuity of x, not y.
Meant to say not y^-1* at the end

>>16769337
bump. is there an advantage to doing differential geometry this way?

>>16770936
Most subjects ought to be taught in a way that has something like a history lesson somewhere in it. Even if some of this is familiar from like Precalculus, the historical progression and the people behind it is great and understanding why things occurred the way it did seems pretty important. Also fun too. More like a movie than a lesson with too much thinking.

>>16770981
im going to bump because im a bastard and wanting to know more details about this homeomorphism business

im a bit torn like i very much appreciate a historical POV on a subject but sometimes it just isn't a good method to learn compared to tight modern theory. some math professors i have seem far too concerned on like the history of a subject or names of things and sometimes its like please get back to the topic.

>>16770936
I wouldn't think so. Continuity is one of the key properties that allows diffeomorphisms and reparameterizations to work. Once you throw out continuity, you might as well throw out the majority of more abstract diff geo.

>>16771422
I think you are replying to the wrong person.

>>16769118 (OP)
I had a book by this author for riemannian geometry, didnt like it

>>16771433
You were asking about Wildeberger's approach to differential geometry on the rationals, right? That's what he builds up in that video (if I'm remembering correctly, it's been a few years), differential geometry on the rationals and without continuity assumptions.

>>16771414
If you're asking about Wildbergers lectures, go to this video of his: https://www.youtube.com/watch?v=WCwoCFdjUcE "Differential Geometry 25" in which he talks about this "charts" approach to manifolds.

But if you're talking about history vs "get to the point" for manifolds as in here, I think it's very useful for examples and intuition to look at algebraic curves to see what were the first "examples" of manifolds people had in mind. In French people call manifolds "topological varieties" and the zero sets of polynonial equations "algebraic varieties" for a reason.

Anyways, there are 2 main approaches to manifiolds: either embed them in Euclidean space (say, as the zero set of some polynomial equations) or either use this "abstract manifold" chart approach. By Whitney's embedding theorem any abstract smooth manifold can be embedded, so these agree for smooth manifolds.

>>16771456
Disclaimer: I'm giving a bit of a false history of manifolds here, go read the wikipedia page or a book on the topic lol.

Now, the cool thing about manifolds is that there was this obvious way of getting "'curvy surfaces that locally look like Euclidean space" by embedding them into Euclidean space, e.g. the surface of a sphere as a manifold of positive constant curvature. It was precisely things like these: curvature, that people realized where intrinsice to the manifold itself and not quite its embedding (you do need a notion of distance, though!). Later on some other invariants (w.r.t. embedding) came to be, like Euler characteristics and orientability, so people were realizing that manifolds should have some intrinsic way of being, and not ""just" their embedding. Thus the "charts" definition of "abstract manifolds"

But what this charts definition means is precisely something that you need topology for: your manifold is going to be covered by these sets (open sets) in which things look locally like Euclidean space of dimension n, for a manifold of dimension n. A way I like to think about this is that you're taking a bunc of open sets of Euclidean space and "gluing them together." (e.g. glue 2 disks to get the surface of a sphere). But if you want to do this "gluing" in a way that really keeps things "locally like Euclidean space" you need to do this in a way that respects the toplogy of Euclidean space, and it has to be done by quotienting by a homeomorphism, i.e. we need continuos map and its inverse to go from one chart to another.

Hope this clarifies things more than not.

>>16770518
https://math.stackexchange.com/a/1357362

>>16770636
>what is mentioned a page later
in the pic >>16770518

>>16771414
>I wanna know more about details of homeomorphism
This thread isn't doing it justice. Just read the first half of his chapter 2, or read the first quarter of the chapter and skim the second quarter. He is somewhat very explicit in wanting to express why things are defined the way they are, and you get the real gist of it if you ignore the proofs and just believe it as true. You can get through it in like 30 min probably by reading and ignoring proofs and practice problems.

>>16771464
I have heard many times about complex manifolds, are these just like euclidean manifolds for twice the dimensions or fundamentally different. I dont know how something complex can look locally euclidean.
Speaking of euclidean, minkowsky space isnt euclidean, in its most banal flat form. Can these things be reduced to euclidean space locally always?

>>16771472
Oh, you're certainly right. I see now. Ill write up in full when im at computer.

Definitely a valid criticism, why the remark doesn't mention the key feature here, idk.

This sort of reminds me of a fairly egregious proof I saw in otherwise one of my favorite topology books, Topology by Hocking and Young. Ill share it when im on computer.

>>16771483
'Locally euclidean' just means 'looks like R^n', nothing to do with curvature.

Confusingly, euclidean can refer to euclidean metric which gives a flat structured manifold.

Minkowski space is locally euclidean but doesn't use a euclidean metric.

Complex manifolds are locally euclidean in the obvious way: a complex plane looks like R^2. Again this has little to do with metric or curvature structure.

>>16771476
Ive read and worked through the book, the discussion is more of its a good book and I find this point of lacking the use of continuous inverse in the proof quite interesting.

Ignoring proofs and practice problems feels like a mistake.

>>16771524
So complex manifolds look locally like R2 or can they have more dimensions?
What does locally euclidean mean if minkowsky space doesnt have an euclidean metric, yet is called locally euclidean?
Just want a definition for "locally euclidean"

>>16771620
NTA, but complex manifolds don't just look locally Euclidean, they essentially are Euclidean.

The whole point of studying holomorphic functions on CN is that they can be locally represented by linear transformations on R(2N).

Also, "locally Euclidean" doesn't require a metric. All it means for a manifold M to be locally Euclidean is that there is some N <= dim(M) such that smooth functions on M can be arbitrarily well approximated by smooth functions on RN. A manifold being locally Euclidean does not necessarily mean that there is some Euclidean metric which can be well defined for that space.

>>16771620
>>16771647

Also, I forgot an important detail. Minkowski space is a pseudo-Riemannian manifold. It has a "metric" in the sense of a positive semi-definite bilinear form on R4, such that any two points can be compared via their projections onto a tangent space using a standard x^T P_q y kind of expression.

This expression is well-defined everywhere in Minkowski space, so in most places you're able to "metricize" up to a set of measure zero. The problem is that the Minkowski metric is positive semi-definite, not positive definite. The metric has a "null-space."

>>16771653
Oops. Ignore this:
> The problem is that the Minkowski metric is positive semi-definite, not positive definite. The metric has a "null-space."

It's not definite. It's relaxed not in the null-space sense, but in the sense of always having an eigenvector with an opposite polarity eigenvalue (e.g., 3 positive and 1 negative or 1 positive and 3 negative).

>>16771443
>>16771456
I was asking about differential geometry on the rationals.

>>16771732
I wouldn't know what you lose or gain, feel free to find out

>>16771472
It kind of reminds me of this. I otherwise love this book but this is one of the most egregiously wrong proofs I've seen: the idea is to show that the composition of a path and its path inverse is homotopic to identity.

I think what kind of happens is an author 'knows' a result to be true and goes through the motions of the immediate idea that comes to mind when they verify it. They are less critical about whether the proof is accurate since well the result is so obviously true. I kind of feel that is what is going on here with do Carmo.

This to me actually showcases the importance of rigor, ie even if you explain the basic idea and handwave it, may be totally wrong since you've missed some details.

>>16769160
I've never seen a book criticized of non-rigorous for having too many examples, the real problem with such books is that they often really put too much detail into an example which won't be useful in a proof, and only sketch those.

>>16769596
>the authors make small mistakes everywhere
Really? Can you give me an example of that?
I guess that by "rigorous authors" you are thinking perhaps of Lang, Bourbaki and/or Serre; but honestly they have little to no errata on their books, and if so they are very well documented.
Generally books with "two many errata" I've encountered are the most new ones, and the authors have always replied my emails back if I don't understand something which was typed wrongly.

>>16770275
>Do Carmo's is a massive larper with an inferiority complex.
I have no clue how do you got into that conclusion. I agree with other posters into that do Carmo is hardly the "hard and rigorous" presentation of differential topology, but if you thought it was not for you, then switch to another like Lee or something, like why so much hatred for the guy?

>>16771483
>are complex manifolds just like euclidean manifolds for twice the dimensions or fundamentally different
Almost. Instead of asking for them to look locally like $\mathbb{R}^n$ we ask them to look like $\mathbb{C}^n$ which is topologically $\mathbb{R}^{2n}$, and to satisfy that the transition maps are analytic/holomorphic. Meaning that after projecting to a single coordinate, they admit a local power series expansion everywhere.

>>16771647
>The whole point of studying holomorphic functions on CN is that they can be locally represented by linear transformations on R(2N).
Strongly disagree. While holomorphic => real differentiable, there are a ton of caveats in complex analysis; essentially everything that can go right, does so. The analytic structure is *very* rigid, as for instance, the set of analytic automorphisms of the upper half plane is a real manifold (impressive, as the differentiable automorphisms of the plane has cardinality greater than the continuum one).

>>16772106
Maybe I'm just retarded, but I don't understand your disagreement. I understand there's a lot of caveats in complex analysis (and I also understand that my 1 semester's worth of grad level complex barely even skimmed the surface of them).

I don't see how that negates the idea that functions defined on complex manifolds "in" C^n are exactly representable by functions on embedded manifolds in R^2n. Isn't that the point of the Whitney immersion theorem?

Can you try again, but with different words? Maybe it will stick the second time around.

>>16772094
>and the authors have always replied my emails back if I don't understand something which was typed wrongly.
Yes, this is very rational. You have to have correspondence with an author because he cant write. Anything for rigor amirite?

>>16771472
Okay I figured it out.

There's another more obvious error here: that h|N =/= F^-1y|N. I mean think about it: h is (subsets of) R^2 -> R^2, F^-1y is (subsets of) R^2 -> R^3. So these can't be equal as functions, but this is easily remedied by stating h x {0}|N =/= F^-1y|N

For the actual issue, all we have is a neighborhood M of x(q) in R^3, then N an open subset of R^2 so that y(N)⊂M. However, while such an N exists, there need to be no guarantee that it is a subset of y^-1(W): the existence of such an N can only be guaranteed if y^-1(W) is *open*, which is guaranteed when x and y inverse are continuous. That is where we use the properties of x and y being homeomorphisms.

The equality h x {0}|N=(F^-1y)|N obviously requires N to be a subset of y^-1(W) since well that is where h is defined and N needs to be open for use this fact to conclude h is differentiable (as differentiability is defined with respect to open sets). So where the proof breaks down without x and y invertible is that there may not exist any such open N contained in y^-1(W) as y^-1(W) may not even be open (for instance, it could be a line in R^2)

Also Proposition 4 likewise is incorrectly stated.

You have a great eye anon.

>>16772252
meant to say "where the proof breaks down without x and y inverse continuous"

>>16772252
I don't get how that is 'rigors' issue. If anything, it is the lack of rigor that leads to wrong results, its when stuff is 'implicitly assumed or taken as granted' that stuff falls through the cracks.

I think the sense you mean though is that rigor can 'obfuscate' when a wrong result is wrong and this can be better cleared up through explanation. That's an issue of clarity in presentation though, not inherently 'rigor'.

I also think this anon is simply pointing out that the authors of these books of many typos aren't negligent and you don't need to be 'out for them'. Writing math books is just hard.

>>16772241
NTA but I think the key point is that of the differentiability requirements: complex manifolds require holomorphic transition maps which is a much much more stringent condition than any form of real differentiability, which creates the key difference in complex and real manifolds.

And why in a complex manifold you do need to show complex charts so that holomorphic transition maps can be defined.

>>16772298
Its fake rigor.
Normal speech can convey complex concepts thorough examples and repetition. Rigor is fine if its not fake

>>16772294
I think this has been addressed at least twice in the thread, and once by me here
>>16770631
oh I think you or someone like you answered here
>>16770638
so let me address that real quick:

If you needed continuity of y to prove that h is differentiable, you need continuity of y^-1 to prove that h^-1 is differentiable. Please write the first 2 or so paragraphs of Do Carmo's proof of Proposition 1 but for h^-1 and see for yourself.

>>16772399
Can you name math book titles that are rigorous without containing fake rigor? Or is this better for another thread?

The citations of this book are weak and it needs to be revised.

When will you realize that the ones who went to the sun are alpha Chad and the moon men are beta

And even worse is when they cannot compete that. Leaving their heir behind... Idiots

>>16772402
Check the detailed response I gave above, that is actually where the proof breaks down.

h=x^-1 y so h^-1=y^-1 x. If neither x and y inverse being continuous was used in proof of differentiability of h, by symmetry h^-1 wouldn't either, h^-1 would just be doing proof for h with x and y swapped.

If you still think continuity of x inverse and y inverse continuity would show up, please point where.

>>16769118 (OP)
whole education system is gaslighting.
Looking back I can say it couldve been taught better.
Looking forward, if you dont get it you are insulted & blamed.

> Thoughts on this book?
Idk never read it.

>>16769210
>let's go right into what matters.
examples & explanations build credibility.
If all you want are end answers you are implicitly assuming authors correctness. This is explicitly false, not even accounting for misunderstandings.

Die alone.

>>16772399
I mean not just examples and repetition, that may have worked in calculus but when things get more abstract you need a better language for stuff.

Like how would you do functional analysis and that kind of stuff without rigor? I was going to say algebraic topology but I suppose some areas are less rigorous.

>>16772301
I really need to brush up on topology. I have a feeling that my first pass at Munkres didn't leave me with as much understanding as it should have.

>>16773110
I'm a weird person that didn't start with Munkres so I remember my thought when I had to use it for class being 'wow, I hate this'. It felt very unmotivated, not even bothering to explain where topological spaces come from. That said, I did grow to appreciate it at least for giving an outline on how to work the 'abstract topologies' in a concise way. Nevertheless, if Munkres was my introduction, I'd have hated the subject. (Hocking and Young) and Willard make point-set topology way more interesting.

But, if you're learning topology you probably really want to learn about stuff more like surfaces and properties for topological spaces in a concise way. I think Janich's book might be good for that. There's also Gamelin and Greene's book which I personally don't like its 'barebones' tackling of well known topological theorems, but it does focus a lot more on actual topological problems you hear about: pancake theorem, ham and sandwich theorem, hairy ball theorem, jordan-curve theorem, etc.

>>16772301
Yep, this was my point.

>>16772241
Funny you mention Whitney, as the classical differential topology proof fails for complex manifolds! The problem is that the classical proof uses partitions of unity which are easily shown to be of class C^\infty, but are never holomorphic as they vanish in a non-discrete subset of a chart. The result is still somewhat true, but as you can see, it is a radically different result in spirit, though it may look alike.

>>16773167
Hmm, you've given me something to think about. Maybe I'll get around to studying complex manifolds specifically.

Do you think it's worth it to study Algebraic geometry/topology first? My algebra background is nowhere near as strong as my analysis background.

I have never met a mathematician who wasn't autistic

imagine not being able to proofmax. skill issue, getting mogged by differential geometry is the biggest L(aTeX) I've ever seen.

>>16770477
kek

>>16773157
I have no formal math education beyond a basic calculus sequence, some linear dynamical systems, a master's level complex course and two master's level real analysis courses. I study this stuff because I find it interesting and every once in a while it comes in handy to solve an engineering problem relevant to me (e.g., just recently submitted a smooth manifolds based signal processing paper to a journal, finger's crossed on the review process going smoothly).

Thank you for the recommendations. I'll try to find some of the books you're talking about.