Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\mathbb{P}^n$:
(Most standard characterisation?) $\mathbb{P}^n$ is the parameter space of complex lines through the origin in $\mathbb{C}^{n+1}$.
(Kobayashi-Ochiai). $\mathbb{P}^n$ is the unique compact Kähler manifold $X$ whose first Chern class is $\lambda c_1(\mathscr{F})$, where $\lambda \geq 1+\dim_{\mathbb{C}}(X)$ and $\mathscr{F}$ is a positive line bundle on $X$.
(Mori, Siu-Yau). $\mathbb{P}^n$ is the unique compact Kähler manifold $X$ with a Kähler metric of positive bisectional curvature (this, in fact, characterises $\mathbb{P}^n$ with the Fubini-Study metric, since the biholomorphism here is also an isometry.
In contrast with spheres, $\mathbb{P}^n$ does not appear to be characterised by the sectional curvature of a Riemannian metric on it. Indeed, by the well-known $\frac{1}{4}$-pinching theorem, a compact Riemannian manifold with a Riemannian metric whose sectional curvature $K$ is pinched $\frac{1}{4} < K \leq 1$ is diffeomorphic to a sphere.
Of course, the Fubini-Study metric on $\mathbb{P}^n$ has sectional curvature pinched by $\frac{1}{4} \leq K \leq 1$, but this does characterise $\mathbb{P}^n$ at all: spheres, the Quaternionic projective space, and the Fake Cayley plane also possess metrics with sectional curvature pinched between $\frac{1}{4} \leq K \leq 1$.
I'm very interested in the following question:
What other characterisations of $\mathbb{P}^n$ exist in the literature?
Additionally:
In particular, are there purely Riemannian characterisations of $\mathbb{P}^n$?
