Timeline for How many ways are there to characterise $\mathbb{P}^n$?
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| Dec 31, 2021 at 18:15 | comment | added | Nick L | If the Blaschke Conjecture is true then the only Riemannian manifolds satisfying this condition also having an almost complex structure (without any compatibility condition with the metric) are $\mathbb{CP}^n$ (with the Fubini-Study metric) and $S^6$ (with the round metric). If one requires that the metric is Kahler then the exceptional case $S^6$ dissapears. The Blaschke Conjecture is known up to diffeomorphism for complex projective spaces and up to isometry for spheres. Searching "Blaschke Conjecture " should yield the relevant references, | |
| Dec 31, 2021 at 18:13 | comment | added | Nick L | For a somewhat different Riemannian characterisation the Blaschke Conjecture may interest you. Definition: A Riemannian manifold of diameter $d$ is called Blaschke if every geodisic from any point of length less than $d$ is the unique shortest path between any two points on it. The Blaschke conjecture states that a manifold satisfying the Blaschke condition is isometric to a compact rank 1 symmetric space. | |
| Dec 30, 2021 at 9:52 | comment | added | user347489 | It seems like the Seshadri constant of the anticanonical at a smooth point also gives a characterization: link.springer.com/article/10.1007/s00209-017-1941-9 | |
| Dec 30, 2021 at 7:51 | comment | added | abx | Though you seem to be more interested by differential-geometric characterizations, let me mention that there are many such characterizations in algebraic geometry: see Cohomological characterizations of projective spaces and hyperquadrics by Araujo-Druel-Kovács (Invent. Math. 174 (2008), no. 2, 233–253), and in particular the introduction. | |
| Dec 30, 2021 at 2:46 | comment | added | Jason Starr | There is also the characterization by Cho, Miyaoka, and Shepherd-Barron: the unique Fano $n$-fold with Fano pseudoindex equal to $n+1$. | |
| Dec 30, 2021 at 1:06 | comment | added | Ryan Budney | Why isn't this a purely Riemannian characterization? Take the canonical circle bundle over the manifold that kills the $2^{nd}$ homotopy group, this is an odd-dimensional sphere and the projective space has the Riemann metric of that quotient space. This is just the higher version of talking about covering spaces of manifolds with $+1$ sectional curvature. You've just replaced covering spaces (i.e. discrete bundles) with circle bundles. | |
| Dec 30, 2021 at 0:05 | history | asked | AmorFati | CC BY-SA 4.0 |