uses of Riemannian geometry for questions not related to Riemannian geometry

Question

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using considerations from Riemannian geometry.

In Wikipedia (https://en.wikipedia.org/wiki/Riemannian_geometry) one finds a long list of results related to Riemannian geometry, however for these results usually either the assumptions or the conclusion of the theorem use notions from Riemannian geometry.

What other questions, besides Poincaré conjecture, whose formulation does not refer to notions from Riemannian geometry, were solved applying, as one of the means, Riemannian geometry considerations?

EDIT I have added the formulation of Poincaré conjecture after a comment which questioned the claim that Poincaré conjecture did not involve Riemannian geometry.

EDIT I tried to make the formulation of the question clearer, and to narrow its scope omitting the "metaconsiderations" part.

Answer 1

For 1, positive mass theorem can be one of the examples. Here is some information about positive mass theorem. Schoen and Yau used the minimal surface to prove the positive mass theorem, which is originally a problem in general relativity.

Answer 2

Here are two suggestions:

1. Morse theory is a central tool in analyzing the topological structure of manifolds. A key step in the theory is studying the gradient flow of a Morse function, which requires a Riemannian metric to define.
2. Donaldson's theorem says that if $M$ is a smooth, simply connected, compact $4$-manifold with definite intersection form, then the intersection form of $M$ can be diagonlized over the integers. Since many simply connected topological $4$-manifolds have non-diagonalizable definite intersection forms, this leads to the conclusion that such $4$-manifolds have no smooth structures. This purely topological theorem was proved by studying the space of instantons on the manifold, which require a Riemannian metric to define.

Answer 3

Here is a simple example: the fundamental group $\pi_1 S$ of a closed oriented surface $S$ of genus $\ge 2$ is a word hyperbolic group in the sense of Gromov.

The outline of the proof goes like this. $S$ has a Riemannian metric of constant curvature $-1$. Its universal covering space is therefore a complete, simply connected Riemannian surface of constant curvature $-1$. Every such space is isometric to the hyperbolic plane $\mathbb{H}^2$. The group $\pi_1 S$ therefore acts on $\mathbb{H}^2$ by a cocompact, isometric deck transformation action. By the theorem of Milnor and Svarc, the group $\pi_1 S$ with its (finitely generated) word metric is quasi-isometric to $\mathbb{H}^2$. The metric space $\mathbb{H}^2$ is a Gromov hyperbolic space. Gromov hyperbolicity is a quasi-isometry invariant, and therefore $\pi_1 S$ with its word metric is a Gromov hyperbolic space. By definition, that means $\pi_1 S$ is word hyperbolic.

Indeed the entire study of word hyperbolic groups was motivated by the study of complete Riemannian manifolds with negative sectional curvature bounded away from zero. This study is a hugely successful enterprise which obtains group theoretic analogues of theorems about negatively curved manifolds.