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The Laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. There is also a Hessian comparison theorem, and versions for different curvature bounds.

These are covered in several books on differential geometry, but I haven't been able to find any information on their history. Where were they first established?

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    $\begingroup$ It seems this goes back to Calabi's "An extension of E. Hopf’s maximum principle with an application to Riemannian geometry". Duke Math. J., 25:45–56, 1958. See projecteuclid.org/journals/duke-mathematical-journal/volume-25/…. $\endgroup$
    – Igor Belegradek
    Commented Nov 30, 2021 at 1:24
  • $\begingroup$ Thanks, that looks like a great paper. I see there's also an extension of the usual Hopf-Rinow theorem. Although that does answer the version of the theorem I put in my question, I am still curious about the more general version. $\endgroup$
    – Quarto Bendir
    Commented Nov 30, 2021 at 1:36
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    $\begingroup$ Modern history can be traced from the references on p.1 of Eschenburg's "Comparison Theorems in Riemannian Geometry", see math.toronto.edu/~vtk/eschenburg-comparison.pdf. I think the subject originates from ODE comparison theorems, see en.wikipedia.org/wiki/Comparison_theorem. For example, Sturm's comparison theorem is in J. Math. Pures Appl. 1 (1836), 106–186. $\endgroup$
    – Igor Belegradek
    Commented Nov 30, 2021 at 2:25
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    $\begingroup$ @IgorBelegradek, I forgot about Rauch. To my surprise, few people cite Rauch's original paper. It appears to be this one: jstor.org/stable/1969309 $\endgroup$
    – Deane Yang
    Commented Dec 1, 2021 at 3:57
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    $\begingroup$ @DeaneYang Bishop announced his work in the 1963 Notices of the AMS $\endgroup$
    – Quarto Bendir
    Commented Dec 5, 2021 at 6:28

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