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Edit : (I didn't intend this as an insult or a debate discussing which way is best or better for what, I'm just asking a question for my interest and I believe in the interest of science, at least for variety sake.. I do not not idealise any man or work, the only reason I brought up principia is to save myself the trouble of answering unending strings of questions on how will I practically calculate without a basis, so that's why I called upon the highest authority in this regard.

I know coordinates are useful when used right, I only have a problem when people say you must use them in practical calculations and it can't be done other way. Invariant formulations are most useful in the long run, when it comes to unification of different areas, and attacking the deepest problems that almost always require some level of unification. If someone is genuinely interested in the details especially for research purposes I can elaborate on this further .)

Basically without pasting any non existant (non intrinsic) structure on an actual space, which for euclidian geometry is an euclidian affine space of points. .

The way they did geometry from the ancient Greeks to Descartes.

Coordinates and their maps are the foundation of standard differential geometry. The theory is coordinate free, but riddled with non geometric objects, and with the need to prove that geometrical objects are not just coordinate nonsense.

I am looking for a theory including differential operators that builds directly on the pre Descartes approach to geometry.

Newton developed the entire principia mathematica this way, and I believe he could have used calculus with that geometric approach.

Is there any such exposition that would deal with differential operators like like covariant derivative, vector fields and differential forms, without assuming any analytical (coordinate) geometry

Matko
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    Newton assumes that the space he works in is Euclidean, giving him symmetries and uniquely determined operators with those symmetries. If you want to work with something less symmetrical, you need a way to grab hold of its invariant differential operators, something which, it appears, must involve heavier machinery. – Ben McKay Dec 21 '20 at 11:24
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    I agree, this is exactly what appears to be the case. – Matko Dec 21 '20 at 11:28
  • But maybe appearances are in some way deceiving, or the machinery could be more canonical – Matko Dec 21 '20 at 11:29
  • I don't see why newtons calculations couldnt be more polished to formalise analogous calculus concepts, and then generalized to the case where the fifth postulate doesn't hold – Matko Dec 21 '20 at 11:32
  • Why the dislikes? – Matko Dec 21 '20 at 12:22
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    For a coordinate-free treatment of différentiel calculus see for instance Henri Cartan's Cours de calcul différentiel – Pietro Majer Dec 21 '20 at 13:30
  • I don't speak French, and I'm not looking just for a coordinate free text, coordinate less text, there is a difference – Matko Dec 21 '20 at 13:32
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    What's the difference between "coordinate free" and "coordinate less"? – Deane Yang Dec 21 '20 at 13:43
  • If you describe a geometric figure by introducing variables that denote some properties of the figure (like arc lengths, areas, distances between special points etc.) would you then be doing coordinate less geometry? I'm not an expert in pre Descarte geometry, but I do believe they introduced variables this way. – Michael Bächtold Dec 21 '20 at 13:46
  • The difference is in that while basis free and coordinate free might only imply independence in regards to a particular choice of basis, frames or coordinates, coordinate less means explicitly(manifestly) independent of coordinates, simply because no coordinates are used – Matko Dec 21 '20 at 13:52
  • @michael yes that is what I consider coordinate less, it was the only geometry available for thousands of years, from the time of Aristotle to the time of Newton, up until descartes – Matko Dec 21 '20 at 13:54
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    I don't see the difference between coordinate-less and variable-less geometry. I doubt the second one is possible, hence to my mind also the first one. – Michael Bächtold Dec 21 '20 at 13:56
  • I don't know what you mean, ever looked at principia, or even for example vector calculus, as long as you don't express vectors as triples of scalar, you can still prove and calculate stuff and have variables without coordinates – Matko Dec 21 '20 at 14:01
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    And here I thought the old-school way was avoiding topoi... – Faris Dec 21 '20 at 14:01
  • Coordinates are variables. In which sense are these different concepts? – Michael Bächtold Dec 21 '20 at 14:10
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    By the way, there is a reason why we all abandoned Euclid's approach to geometry and prefer working with Cartesian coordinates. It's a lot easier to prove theorems using the latter, even with the additional step of proving coordinate independence. – Deane Yang Dec 21 '20 at 14:20
  • I'm not here to debate which is better, I'm just asking for an alternative to coordinates, that's all – Matko Dec 21 '20 at 14:48
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    related: https://mathoverflow.net/questions/14877/how-much-of-differential-geometry-can-be-developed-entirely-without-atlases https://math.stackexchange.com/questions/53021/defining-a-manifold-without-reference-to-the-reals –  Dec 21 '20 at 15:11
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    I find I can persuade myself that coordinates aren't ugly if instead of thinking of a vector in $\mathbb{R}^n$ as a tuple of coordinates I regard it as an intrinsic geometry object that (alas) has to be represented in this way after a choice of basis. So really a vector is an equivalence class of pairs (basis, coordinates), one transforming contravariantly to the other, and no single coordinate system has to be preferred. I'm no expert, but I think much the same works replacing 'vector' with 'chart'. – Mark Wildon Dec 21 '20 at 15:31
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    I will shut this question down if I get one more comment on the justification of coordinates. I am not saying coordinates are bad, so you don't need to defend yourself. Maybe I phrased the question wrong. But you're right mark charts do give manifolds their algebraic structure like vectors, I'm just looking for a source that does it in a diffent way. I'm not condemning anybody – Matko Dec 21 '20 at 15:42
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    I have to admit that I share some of your views. When I teach differential geometry, I do like to start by recalling what classical Euclidean geometry is, as well as the need to develop coordinate-independent physical laws, and emphasizing that differential geometry has the same ultimate goal but for curved spaces. And I do believe that often a more abstract approach is easier to understand than using coordinates. But in practice when you're actually trying to prove theorems or do calculations, coordinates or moving frames are almost always easier. – Deane Yang Dec 21 '20 at 16:38
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    What people like Burago, Ivanov, Petrunin, and others are doing with metric geometry is very much in the spirit of how Euclide did geometry, but it requires a much deeper geometric insight than what I have. As others have said, you might want to investigate their work. If you can understand it, I believe it's quite beautiful. – Deane Yang Dec 21 '20 at 16:41
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    I'm fairly confident you can do linear algebra, multivariable calculus, differential geometry etc without ever choosing a basis (so "coordinate free"). On the other hand this is very far from what Euclid would do, of course. Would that satisfy you? – Denis Nardin Dec 21 '20 at 18:30
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    There have been several very similar questions to this one in recent years (coming from various user accounts), with an equal insistence that the questioner wants to do diff geom or calculus without mentioning coordinates at all, which is a much stronger requirement than demonstrating "coordinate independence". It might help if the present account wrote out an explicit indication or example of something in calculus which is defined-and-proved without using coordinates, rather than just telling respondents that what they propose does not meet the requirements. – Yemon Choi Dec 21 '20 at 18:46
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    That is: the present account keeps referring to Newton's Principia, but surprisingly enough I do not have a copy on my bookshelf or my hard drive. So constantly saying "look Newton did calculus without Cartesian coordinates so there should be diff geom without mentioning coordinates" still leaves it unclear for many readers what the OP precisely wants. (What kind of definition of "3-dimensional Euclidean space" can one use, for instance? Is one allowed to mention the notion of a basis for a vector space?) – Yemon Choi Dec 21 '20 at 18:49
  • BTW there is a definition of the derivative for a function f: U \to E, where U is an open subset of a normed vector space X and E is another normed vector space, which never mentions partial derivatives/Jacobians etc. (I know many in this comment thread know this but I wonder if it is the kind of thing which the OP is seeking or finds acceptable.) – Yemon Choi Dec 21 '20 at 18:53
  • @Deane Yang I asked this question to see what the current state of research is in terms of invariant geometry. It is not a matter of one over the other. In everyday math work is is easier and more practical to use whatever particular structure is convinient,instead on delving too deep on the very fabric of foundations, but in a more global view, when it comes to bridging different fields and tackling the biggest problems invariant reasoning is the most effective and practical. The reason I brought up Newton is to demistify what it means to be coordinate less. Look at his brachistochrone proof – Matko Dec 21 '20 at 20:17
  • No, obviously from the description, a basis is a big no no, as it is the most obvious example of pasting non canonical structure. Again..., I'm not saying this is bad in general but simply as I explained it should not be the only way – Matko Dec 21 '20 at 20:20
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    I am thankful for the interesting links and books Iv got, and, if anyone is interested, first time I fell through this rabbit hole is due to baez and his gauge fields knots and gravity. There is also misners gravitation. Two of the most beautiful books Iv come across ever.. And both emphasize the synergy of coordinate and coordinate free approaches. I have just found that, athouh usually much harder and more impractical, purely invariant ways have their merits, that's all – Matko Dec 21 '20 at 20:25
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    if you do not allow yourself the definition of a basis then what, pray, is your definition of finite-dimensional? How do you even know that you are working in the Euclidean plane as opposed to Euclidean 3-space? It seems to me that you are attracted by a dream of synthetic geometry but are being very dogmatic about what should be allowed in it – Yemon Choi Dec 21 '20 at 21:56
  • Also, why all the different user accounts? – Yemon Choi Dec 21 '20 at 21:57
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    'when it comes to bridging different fields and tackling the biggest problems invariant reasoning is the most effective and practical" - citation needed. You are on a site which is read by and commented upon by research-level mathematicians, and yet you repeatedly make sweeping assertions like this without really backing them up. What is your actual mathematical experience (not physics!) which supports this? – Yemon Choi Dec 21 '20 at 22:08
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    By the way, a trick I learned from Robert Bryant is that, if you do your calculations on the orthonormal frame bundle (or sometimes on the full not-necessarily-orthonormal frame bundle), the all of the differential forms are defined globally and you do not have to prove invariance under change of frame. – Deane Yang Dec 21 '20 at 23:14
  • Keep your basis tricks for yourself. I simply asked for a reference, I know what I want, and by some miracle I got a few good references and a nice answer. I don't wish to preach invariant philosophy and praxis nor am I against non invariant ways. Only problem I have with it is that people say it's that way or the highway, you can't do anything without a basis, yada yada.. As I said both ways have their drawbacks and benefits. I could wrote a book about c.free formulations and their merits, and especially on how exactly they leat to new advances and unifications. – Matko Dec 22 '20 at 09:08
  • "Modern work, however, has shown the value of coordinate-free geometric formulations. Likewise there is also a value in geometrically invariant derivations. Not only do they directly show that the result is coordinate independent they also serve to clarify certain assumptions." https://www.google.com/url?sa=t&source=web&rct=j&url=http://citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.485.7856%26rep%3Drep1%26type%3Dpdf&ved=2ahUKEwj3xdejnuHtAhXvs4sKHfRiCXoQFjABegQICxAB&usg=AOvVaw13xVUr_G8OswNiWUE3F1L0&cshid=1608628306199 – Matko Dec 22 '20 at 09:12
  • I do not idealise any man or work, the only reason I brought up principia is to save myself the trouble of answering unending strings of questions on how will I practically calculate without a basis, so that's why I called upon the highest authority on that matter. – Matko Dec 22 '20 at 09:18
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    "I could wrote a book about c.free formulations and their merits, and especially on how exactly they leat to new advances and unifications" -- this sounds like a much more productive activity than constantly claiming research mathematicians here have some kind of deficient understanding – Yemon Choi Dec 22 '20 at 20:48
  • But if my claim is right, although I didn't exactly explicitly say what you imply, who will care about that, unless I actually do some great breakthrough or unification, and who will care to check it... Thaugh you do have a point.. – Matko Dec 22 '20 at 21:16
  • Other readers coming late to this long exchange of comments might wish to check out questions on math.SE by the same user and his or her replies to interlocutors: https://math.stackexchange.com/questions/3608556/what-is-a-coordinate-free-definition-of-sine and https://math.stackexchange.com/questions/3956355/is-it-possible-to-do-calculus-and-differential-geometry-the-old-shoool-way-with – Yemon Choi Jan 22 '21 at 21:30

3 Answers3

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The Geometry of Geodesics, by Herbert Busemann, provides a purely intrinsic approach to a large part of differential geometry, through axioms on the metric.

  • It does not define covariant derivatives — but it defines geodesics without them, as length-preserving maps from the real line.

  • It does not define vector fields — but it analyzes motions, which are a finite analog to that infinitesimal notion.

  • It does not define differential forms — but it defines scalar curvature synthetically.

Busemann then proved a whole book of impressive theorems on this basis. (I gave some examples at Characterizations of Euclidean space) If you want a result in Riemannian geometry that you can state without coordinate definitions, you’ll probably find a proof there.

  • This description matches exactly to what I had in mind, I always considered geodesics more fundamental than covariant derivative, dif forms i find quite esthetically pleasing as a concept but couldnt find a way where they are not irreducible to coordinates – Matko Dec 21 '20 at 14:09
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    Henri Cartan and others did find ways to do differential forms without coordinates. But these approaches usually obscure rather than elucidate what's going on. – Deane Yang Dec 21 '20 at 14:18
  • I don't have much access to his work, but he used frames which is pretty much the same thing – Matko Dec 21 '20 at 15:09
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    @ArcDDD, Henri's father Elie used frames. Henri worked on more modern coordinate-free formulations of aspects of differential geometry. – Deane Yang Dec 21 '20 at 15:57
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    There has been a lot of progress since that book was written - you might consider the more recent textbook "A Course on Metric Geometry" by Burago, Burago, and Ivanov. One of the premises of this area is that if you can prove some of the main theorems of Riemannian geometry using just low-level concepts like distances between points, lengths of curves, angles between curves, etc. then you can generalize them to spaces with singularities - this was one of the key perspectives that Perelman brought to the Poincare conjecture, for example. – Paul Siegel Dec 21 '20 at 15:58
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    Other good stuff to read includes the papers of both Buragos, Perelman, Ivanov, and Petrunin - the latter two are active on MO from time to time. – Paul Siegel Dec 21 '20 at 16:00
  • Exactly, it is all about the primitives of distances lenghts and angles, and I find this in the long run to be very practical, as supported by perelman and his proof, I believe through even more painful refinement, even more great results would come simply by reducing the complexity of the underlying machinery – Matko Dec 21 '20 at 16:08
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    Thaugh these books and perelman all used coordinates in some way, and that's probably the best way to achieve what they intended, – Matko Dec 21 '20 at 16:12
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    @PaulSiegel, the OP used the word “intrinsic”, and that’s not the way I would describe an approach having key definitions based on comparisons with Euclidean space. But I agree that the contemporary research like Busemann’s is closer to the Burago-Burago-Ivanov approach. –  Dec 21 '20 at 22:30
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I do think you're asking a reasonable question, but many do not like your way of asking it. It would be better received if you could express it more rigorously and mathematically and showed that you have thought about it more deeply than your wording indicates. After all, this is a research math forum. But let me make some comments.

The first thing is Newton versus Descartes. I have never read Newton's works, so I could be wrong. But since Descartes preceded Newton, I believe Newton must have embraced Cartesian coordinates and used them in his work on planetary motion and the shape of the earth. Is that not so?

As for developing differential geometry without coordinates, many mathematicians, including me, have tried. I'm not sure whether you're talking about surfaces in Euclidean space or abstract spaces known as manifolds. In either case, my impression is that the hardest steps are right at the beginning. First, you need to develop multivariable calculus without coordinates. This can be done but is it worth the pain? Not as far as I can tell, but you can see if you can do it. I definitely could be wrong about that. Second, it's defining what a surface or manifold is.

Some very abstract-minded mathematicians did manage to do this for manifolds, but you lose all geometric intuition and end up in a very algebraic world. Is it worth the pain? Also, not as far as I can tell. After you've defined a manifold, then you can work out the fundamentals of Riemannian geometry using only abstract vector fields. This is demonstrated both in Milnor's monograph Morse Theory and the book by Cheeger and Ebin, Comparison Theorems in Riemannnian Geometry.

As for a surface in Euclidean space, you could first define Euclidean space as an abstract vector space with an inner product. Then you could define a surface to be the level set of a function whose gradient is nonzero and work with derivatives of the function (without using coordinates). The geometry of the surface can now be derived from studying curves in the surface and their derivatives. Some of this is very nice, but some aspects are still easier to calculate and understand using coordinates. In particular, it's difficult to work out examples without using coordinates.

However, in the long run, what professional differential geometers discover is the following: Our main goal is to prove interesting new theorems as efficiently as possible. The most efficient approach depends on the specific circumstances. So we dump the ideology and pragmatically learn how to use all of them. We switch between them as needed. So the fact is that using coordinates is often the easiest way. The basic reason for that is partial derivatives commute. This fact is fundamental and used all the time. Without using coordinates or differential forms (as when using orthonormal frames), that fact is hard to use efficiently.

I do continue to think about all this in the context of teaching differential geometry. I do agree that coordinates can often obscure what's really going on. I don't like most textbooks on elementary differential geometry. So I do try to think of coordinate-free approaches that better elucidate the geometry. Sometimes I succeed. Otherwise, it's coordinates or orthonormal frames. Whatever works best.

Deane Yang
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    Newton did not embrace Cartesian coordinates in his work on planetary motion, as you can see from browsing the Principia. –  Dec 21 '20 at 14:27
  • @MattF., thanks. What about expressing the non-spherical shape of the earth? How did he do that without Cartesian coordinates? – Deane Yang Dec 21 '20 at 14:32
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    If I recall correctly, the Prinicipia argues that the earth is oblate, talking about this in terms of elliptical cross-sections. The more detailed discussions in the 18th century were largely by Frenchmen who did embrace Cartesian coordinates. –  Dec 21 '20 at 14:38
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    @MattF., many thanks for correcting me. – Deane Yang Dec 21 '20 at 14:42
  • I understand all you have said, and I don't quite disagree. You're right in many regards, I just think it is healthy to rewise some things and ask questions, try different ways, and maybe in the long run, in terms if a bigger mathematical picture a more abstract way would prove more fertile, maybe not. It's just a question, I don't wish to insult anybody, especially not people like me who just want to learn more and really love math. But I think abstraction is the key both for understanding and ultimately for practical purposes, in the long run, – Matko Dec 21 '20 at 14:45
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    @ArcDDD, your question and your comment here assume that mathematicians have not tried. That's an unwarranted assumption. – Deane Yang Dec 21 '20 at 14:47
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    @ArcDDD, and if you know this stuff, you should try to do this yourself. A great way to learn differential geometry is to be a little arrogant and try to find a better way yourself. – Deane Yang Dec 21 '20 at 14:49
  • I have tried, and I have done quite well, but why go on to reinwent the wheel? I'm not that arrogant you know.. – Matko Dec 21 '20 at 15:03
  • Provided it has been invented, the coordinate free wheel.. – Matko Dec 21 '20 at 15:04
  • PS you commutation of partials is not a coordinate thing, if you have a function f(r, t), the order of partials shouldn't matter should it? – Matko Dec 21 '20 at 15:15
  • I'm not saying people haven't tried, I don't know if they have, I'm saying people go down the path of least resistance, but I'm not saying that one path is better, I thing both should be avaliable to study. – Matko Dec 21 '20 at 15:16
  • Amen........... – Moishe Kohan Dec 21 '20 at 15:29
  • And I certainly don't think I could do it alone in less than a 100 years, but, quite frankly, I know with a team of specialists in certain areas, I have a program that would produce produce a formulation of differential-geometry that smoothly integrates all its related and sub areas, (cohomology, algebraic topology, complex analysis..) – Matko Dec 21 '20 at 15:32
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    @ArcDDD, I'm not asking you to do it all. But if you have a good start, why not write it up and try to post it on the web and share what you have with others. Then we can build on it. – Deane Yang Dec 21 '20 at 15:50
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    @ArcDDD, the moment you write $f(r,t)$, in my view you are using coordinates named $r$ and $t$. Why is that not the case? – Deane Yang Dec 21 '20 at 15:52
  • Well sometimes it is just a matter of nomenclature and interpretation, but here r could also be a vector – Matko Dec 21 '20 at 16:03
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    If you’re OK with $f(r, t)$ as long as $r$ is a vector (but then what does the partial with respect to $r$ mean?), then you’re OK with coordinatising a high-dimensional space in terms of a lower-dimensional one times $\mathbb R$. But why should $\mathbb R^3 \cong \mathbb R^2 \times \mathbb R$ be OK, but not $\mathbb R^2 \cong \mathbb R \times \mathbb R$? – LSpice Dec 21 '20 at 16:12
  • t is a scalar parametar, r has nothing to RxR, it's an n dimensional vector, partial would be keeping the time fixed and differentiating wrt a vector variable, – Matko Dec 21 '20 at 20:46
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    @ArcDDD, it would be interesting if you would post some of what you've worked out. It would help us better guide you about what's been done in similar directions, if there is any. – Deane Yang Dec 21 '20 at 21:04
  • @DeaneYang I increasingly think that what the OP wants is for a purely synthetic description/axiomatization of concepts in diff geom, and that the talk of "coordinates" is slightly misleading or distracting. Quite why they have such faith that this is feasible let alone desirable is slightly unclear to me, but I can imagine some kind of 3D-version of Euclid's axioms, and then maybe there is a way to talk about a "space" being locally Euclidean, etc – Yemon Choi Dec 21 '20 at 22:02
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    @YemonChoi, as I've said in my post and comments, I agree at least somewhat with OP. And, by now, I think they might know more and have thought about this more than it seemed at first. I definitely prefer approaches to differential geometry, where the geometric invariance is evident at every step and one does not have to prove it as a separate step. And, if you really have to use coordinates, doing it in the least messy way possible. If you look at the bottom of my web page, you'll see some short notes. Some derive geometric invariants using coordinates and some do it by avoiding them. – Deane Yang Dec 21 '20 at 23:11
  • , without dwelling on the intricacies of it all , a geodesic is just a continuous curve that satisfies some local notion of triangle equality wrt the abstract metric definition. My conception of what intrinsic diff geo is, is the same as Busemanns, namely that it's simply the geometry of geodesics. But this the insisting on intrinsic formulation need not be as strong, as I see more and more the importance of embedding theorems and in that In general differential geometry really reduces to ordinary calculus of smooth maps on, open subsets of vector spaces. – Matko Dec 22 '20 at 17:49
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It's possible to do differential geometry in a purely intrinsic way, at least once you've gotten past the initial hurdle of defining what a manifold is. The standard definition of a manifold is a second countable, Hausdorff, locally-Euclidean space, so coordinate charts naturally show up (due to that last part). It might be possible to avoid charts entirely, but it almost requires a new definition for manifold. But once you've gotten past this issue you can do everything else in a coordinate-free way, if you so choose.

The real reason that most geometers fo not do this is that it makes explicit computations extremely difficult. Intrinsic approaches and notation have a philosophical appeal, but are ill-suited for many application, where you might need to compute six or seven derivatives. Picking a convenient coordinate chart (or orthonormal frame) to make the analysis easier is absolutely worth the conceptual loss of simplicity. In fact, there are insights that can be found using a particular choice of coordinates that are nearly impossible to see (or fundamentally more difficult to prove) using a more abstract approach.

Gabe K
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  • That's all fine, but I have never seen such an approach, I am not saying standard approach is bad, it's just frustrating that there is no alternative that I have found which meets the criteria in question completely. – Matko Dec 21 '20 at 15:36
  • I have never seen a non coordinate differentiable manifold definition. – Matko Dec 21 '20 at 15:49
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    It's more or less impossible to define a manifold without mentioning $\mathbb{R}^n$. But here's a definition without using the word "coordinates": https://math.stackexchange.com/a/2134594/10584. – Deane Yang Dec 21 '20 at 15:55
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    Now that you've given more background, I can see that my answer is not really answering your question. However, I still want to emphasize that there are situations where the coordinates (or the frame) are intrinsic geometric objects and not just convenient artifacts that we carry around for calculations. – Gabe K Dec 22 '20 at 15:52