Blaschke conjecture

On a round unit sphere, the shortest route from a point is unique until its antipode, at distance π. That distance is also the largest distance between points. Blaschke’s question asks whether this agreement forces similarly rigid geometry in every dimension.[1] [2]

On a round sphere, shortest paths stay unique until antipodal points, and that stopping distance is the sphere’s diameter. Does the same agreement, measured from every point of a closed connected Riemannian space, force all distances to match one of the standard symmetric models, after a common change of scale? The injectivity radius records how far geodesic coordinates stay unique and nonsingular from a starting point; diameter is the largest distance between points.

Let (M,g) be closed and connected, with injₚ(M,g)=diam(M,g) for every p. The conjecture asserts that (M,g), up to multiplying g by a positive constant, is a standard compact rank-one symmetric space. Equal least periods of geodesics alone—the Zoll condition—is weaker and is not the target.[1] [2]

The round-sphere and real-projective cases are settled. Other projective geometries remain part of the general metric question. Knowing the underlying space’s topology does not show that its distances are exactly the standard ones.[1] [2]

Classical sphere and real-projective metric results leave the unrestricted projective-type rigidity problem. Song’s entire adapted-complex-structure hypothesis permits VMRT recognition and proves the desired isometry in that subclass. It must not be silently assumed for all Blaschke manifolds. Earlier survey corrections also preclude recycling withdrawn topological arguments.[1] [2]

It asks how much shape is determined by shortest paths. An unexpectedly simple distance rule could force a rich global symmetry.[1] [2]

The metric assertion links cut loci and geodesic behavior to algebraic geometry and highly structured symmetry. Success requires recovering the standard metric, beyond recognizing only the manifold’s topological type.[1] [2]