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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

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    $\begingroup$ 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. $\endgroup$
    – Ben McKay
    Commented Dec 21, 2020 at 11:24
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    $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 14:20
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    $\begingroup$ 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 $\endgroup$
    – Matko
    Commented Dec 21, 2020 at 15:42
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    $\begingroup$ 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. $\endgroup$
    – Yemon Choi
    Commented Dec 21, 2020 at 18:46
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    $\begingroup$ 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?) $\endgroup$
    – Yemon Choi
    Commented Dec 21, 2020 at 18:49

3 Answers 3

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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.

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    $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 14:18
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    $\begingroup$ @ArcDDD, Henri's father Elie used frames. Henri worked on more modern coordinate-free formulations of aspects of differential geometry. $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 15:57
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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Dec 21, 2020 at 15:58
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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Dec 21, 2020 at 16:00
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    $\begingroup$ Thaugh these books and perelman all used coordinates in some way, and that's probably the best way to achieve what they intended, $\endgroup$
    – Matko
    Commented Dec 21, 2020 at 16:12
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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.

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    $\begingroup$ Newton did not embrace Cartesian coordinates in his work on planetary motion, as you can see from browsing the Principia. $\endgroup$
    – user44143
    Commented Dec 21, 2020 at 14:27
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    $\begingroup$ 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. $\endgroup$
    – user44143
    Commented Dec 21, 2020 at 14:38
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    $\begingroup$ @ArcDDD, your question and your comment here assume that mathematicians have not tried. That's an unwarranted assumption. $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 14:47
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    $\begingroup$ @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. $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 14:49
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    $\begingroup$ @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? $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 15:52
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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.

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  • $\begingroup$ 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. $\endgroup$
    – Matko
    Commented Dec 21, 2020 at 15:36
  • $\begingroup$ I have never seen a non coordinate differentiable manifold definition. $\endgroup$
    – Matko
    Commented Dec 21, 2020 at 15:49
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    $\begingroup$ 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": math.stackexchange.com/a/2134594/10584. $\endgroup$
    – Deane Yang
    Commented Dec 21, 2020 at 15:55
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    $\begingroup$ 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. $\endgroup$
    – Gabe K
    Commented Dec 22, 2020 at 15:52

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