soft question – What’s the point of differential geometry?

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 semiparametric efficiency theory, Bickel et al., 1998). 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).

There is also a more parametric-focused subfield of statistics called information geometry that leverages euclidean different 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.