I incorrectly meant strong convergence. I am glad that the proof still goes through mostly unchanged with only weak convergence. Thank you, Michael Renardy.

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).

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).