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I've been self studying differential geometry for a little while now (4-6 months). I am learning from Lee's Introduction to Smooth Manifolds, and I just don't quite get the point of the subject. Why do we study the constructions that we study, such as differential forms, submanifolds, vector bundles, etc. ? What is the goal of differential geometry, i.e., what is the motivation for studying it?

Edit: As Will Sawin suggested, I will describe things I find motivating. I find beautiful structures, unsolved problems, and a general goal of what we are trying to do in the subject to be motivating. For instance, the motivation in topology (point-set) is to find topological invariants of a given space. This leads us to connectedness, compactness, the fundamental group, etc. In this example, we have a general rule for what we are trying to accomplish. What is this such rule in differential geometry?

Daniel Waters
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    You might want to read something about its applications in general relativity (perhaps from Weinberg) or classical mechanics (from Arnold). Reading from Lee gives a good introduction to the subject from the inside, but not to its applications. – Ben McKay Jul 18 '21 at 16:31
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    I think you will need to ask a more specific question to get a good answer on this site. At the very least you need to say what kinds of things you find motivating, e.g. applications to other branches of science, applications to other subfields of mathematics, beautiful structures, grand unsolved questions, etc. – Will Sawin Jul 18 '21 at 16:32
  • @WillSawin I edited the question. – Daniel Waters Jul 18 '21 at 16:36
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    The motivation of differential topology is to find invariants of manifolds under diffeomorphism, natural since the tools of calculus and differential equations use derivatives and not just continuity. But then Riemannian metrics provide a means of rigidifying (one of many means) which allows us to use analytic methods to relate global problems in differential topology to local curvature estimates. Similarly Lie groups allow us to use symmetry methods to study differential topology. So we look not just to differential topology but to a broad field of differential geometry. – Ben McKay Jul 18 '21 at 16:50
  • @BenMcKay Thank you for your comment. Perhaps I have spent so long being lost in the details and the tools of differential geometry to really get the bigger picture. – Daniel Waters Jul 18 '21 at 17:06
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    I strongly second Ben McKay’s recommendation. A little basic classical mechanics will go a very long way towards making sense of the basic definitions of differential (especially Riemannian) geometry. I’d also recommend looking at the classical differential geometry of curves and surfaces (as culminating, say, with the Theorema Egregium and Gauß–Bonnet theorem)—a lot of basic differential geometry involves the intrinsic generalisation of conceptually transparent (but superficially extrinsic) constructions on submanifolds (e.g., curves and surfaces) of Euclidean space. – Branimir Ćaćić Jul 18 '21 at 17:09
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    Some "big themes" I have noticed in differential geometry, as a non-expert, beyond what Ben McKay said, are "How can we define, and find, the `best' metric on a given manifold?" and "How can we find invariants of a Riemannian manifold that tell us something interesting about the dynamics of a classical or quantum particle on the manifold"? – Will Sawin Jul 18 '21 at 17:19
  • If you like unsolved problems, this answer by @alvarezpaiva is a good place to start: https://mathoverflow.net/a/137043/44143 –  Jul 18 '21 at 17:39
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    For me it all made scence when I learned about Riemannian Geometry. You want to do all the geometry Gauss made with surface, but "from the inside", living in a manifold that, a priori, is not embedd in the euclidean space (but actually it is because of Whitney theorem). Examples I like are space-time, proyective/grassmanian/flag manifold and the Tangent bundle. A Riemannian metric in this manifolds gives them actual geometry, we can talk about geodescis, volume, distance and many more things. We need differential geometry to define "Riemannian metric" (but is an area of reaserch of its own). – Marcos Martínez Wagner Jun 14 '23 at 06:22

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In Physics, most theories can be formulated in a differential geometric framework:

  1. General relativity: Space-time is modeled as a 4d-pseudo-Riemannian manifold. Frederic Schuller has an excellent set of lectures on this: https://www.youtube.com/watch?v=7G4SqIboeig
  2. Electromagnetism: Has an elegant formulation in terms of differential forms. The generalized Stokes theorem for differentiable manifolds is critical here.
  3. Quantum mechanics and Quantum field theory also have differential geometric formulations. I believe Frederic Schuller speaks more on this in his public Quantum Mechanics lecture series.
  4. Hamiltonian (classical) mechanics models the state-space (momentum + position) as a manifold.

The key benefit of formulating physical theories in terms of an abstract (coordinate-free) manifold is that this ensures that if two scientists in different coordinate frames develop the "same" theory for some physical phenomenon then the two theories will agree with one another (i.e. are equivalent up to a change in coordinates).

In Statistics, the general theory of statistical efficiency theory is built upon (infinite-dimensional) differential geometry (See Bickel, Klaassen, Ritov, Wellner, (1998). Efficient and Adaptive Estimation for Semi Parametric Models. Chicago Journal of Theoretical Computer Science). Specifically, the theory models the statistical model (space of probability distributions) as a Hilbert manifold where the tangent spaces are now Hilbert spaces. The theory then considers the pathwise derivative of smooth functions/parameters of the Hilbert manifold where the pathwise derivative can be viewed as a linear mapping on the tangent spaces. In this case, the statistical model is truly an abstract manifold with charts being the densities of probability distributions that are dominated by some measure $\mu$. Since probability distributions are not dominated by a single measure, there is no global coordinate chart for a fully nonparametric statistical model. The tools of abstract manifold theory allow one to rigorously formulate the notion of a smooth statistical model and develop statistical theory without restricting ourselves to working with densities (which are arbitrary as they depend on the choice of dominating measure). Loosely speaking, this ensures that if two people derive the same statistical theory in the density space but with different dominating measures, we can be confident that their theories will agree. (This is also the motivation in physics). Here is one document I could find that goes over some of the differential geometric formulation of efficiency theory: http://www.stat.columbia.edu/~bodhi/Talks/SPThNotes.pdf. See section 5, especially.

There is also a more parametric-focused subfield of statistics called information geometry that leverages euclidean/finite-dimensional differential geometry. Also, in recent years, manifold learning has become of great interest. For example, in brain-imaging, the brain is modeled as either a 2d or 3d manifold. People have generalized a number of well-known machine learning algorithms to this setting. Differential geometry plays a key role here.

Computer science and optimization. Functional gradient descent and optimization/gradient descent on manifolds are current research areas that leverage geometry. The gradient flow on a manifold is a common object here. See this presentation for instance: https://www.math.cmu.edu/users/slepcev/Chi_squared_SVGD_presentation.pdf. Also, convex geometry/optimization borrows a lot of ideas from differential geometry (e.g. tangent cones).

Personally, my biggest motivation for studying differential geometry was to understand smooth maps on smooth (possibly infinite-dimensional) surfaces. The notions of "paths" on manifolds and derivatives of functions along paths are critical for this. Also, viewing the (path-wise) derivative of a function as a linear mapping on the tangent spaces (and push-forwards and pull-backs) is a powerful abstraction.

Lars
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  • Thank you for answering! Physical motivation might help me to understand more what the constructions mean. As for the abstract mathematical motivation, the comments above addressed that. Thank you very much! – Daniel Waters Jul 18 '21 at 18:44
  • What is the reference cited for the claim about statistics? I do not see that title and year in this list of papers by Bickel: https://www.stat.berkeley.edu/~bickel/bibtex.html –  Jul 18 '21 at 20:01
  • My reference wasn't correct. The actual reference is: Bickel, Klaassen, Ritov, Wellner, (1998). Efficient and Adaptive Estimation for Semi Parametric Models. Chicago Journal of Theoretical Computer Science. As a note, I don't think this book (or other resources for that matter) offer a truly rigorous construction of the statistical model as a Hilbert manifold. But, if you are familiar with the standard manifold construction, then it should be doable to fill in the details. – Lars Jul 18 '21 at 20:13
  • Early work of Aad van der Vaart might also be a good resource (e.g. his book Asymptotic Statistics, 2000). Also any resource on the theory of efficient influence functions/canonical gradients of path-wise differentiable parameters should get into this. – Lars Jul 18 '21 at 20:21
  • I don't see much differential geometry in the table of contents for either of those books. Surely such mathematical authors think in differential-geometric terms sometimes, but that seems too distant to provide motivation for someone learning differential geometry. I like the other paragraphs of your answer, but I think the answer would be stronger without the efficiency theory. –  Jul 18 '21 at 20:53
  • The notion of tangent spaces and linear operators on them are abundant in efficient estimation. Computing the tangent space of a user-defined statistical model as well as the derivative operator of a target parameter/functional and its canonical gradient/efficient influence function are the first steps for constructing nonparametric efficient estimators for a new problem of interest. These concepts are the backbone of the one-step efficient estimator, efficient estimating equations, and targeted maximum likelihood estimation (these methods are commonly used in causal inference). – Lars Jul 18 '21 at 21:50
  • I do agree that this application of differential geometry isn’t as deep as it is for general relativity and physics more generally. But, it is an example of, albeit simple, differential geometric calculations being used in real-world application (like constructing an efficient estimator for the average treatment effect). – Lars Jul 18 '21 at 21:51