The weak Blaschke conjecture for \(\mathbb{C} P^ n\)
type: Item
instance of: scholarly article
title: The weak Blaschke conjecture for \(\mathbb{C} P^ n\)
zbMATH Open document ID: 0817.53029
DOI: 10.1007/BF01232252
author: Alexander G. Reznikov
published in: Inventiones Mathematicae
publication date: 1995-07-20T00:00:00Z
full work available at URL: https://eudml.org/doc/144223
review text: A compact Riemannian manifold \(M\) is called a \(C_{2\pi}\)-manifold, if all its geodesics are closed of (necessarily equal) length \(2\pi\). This class includes (A. Allamigeon and F. Warner) Blaschke manifolds, for which all cut loci \(\text{Cut}(p) \in T_ p M\) \((p \in M)\) are round spheres. The examples are CROSS: a sphere or projective spaces over classical fields \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\), \(Ca\) with standard Riemannian metric and in the latter case \(\dim M \in \{8,16\}\). If \(M\) is a simply connected \(C_{2\pi}\)-manifold, then it is homotopically equivalent to CROSS (R. Bott and H. Samelson). A simply connected Blaschke manifold is actually homeomorphic to its model CROSS (Zbl 0534.53039, Zbl 0557.53023, Zbl 0601.53043, Zbl 0704.57018 and others). The well-known Blaschke conjecture [\textit{H. L. Besse}, Manifolds all of whose geodesics are closed (1978; Zbl 0387.53010); \textit{G. Teumer}, Wiss. Z. Ernst-Moritz-Arndt Univ. Greifsw., Math.-Naturwiss. Reihe 35, No. 3, 62-64 (1986; Zbl 0618.53039)], that any Blaschke manifold is isometric to its model CROSS, had been proved for some partial cases by scheme: 1) by using integral geometry in the space of geodesics one shows that \(\text{vol}(M) \geq \text{vol(model CROSS})\) with equality of volumes if and only if \(M\) is isometric to the model CROSS, (Zbl 0387.53010, Zbl 0725.53046 and others); 2) on the other hand, one uses topological arguments to show that the Weinstein integer \(i(M) = \text{vol}(M)/\text{vol}(S^{\dim M},\text{can})\) is the same as for model CROSS, (Zbl 0387.53010, Zbl 0487.53041, Zbl 0491.53039, Zbl 0595.53048, Zbl 0685.53036, Zbl 0755.53031 and others). The step 2) is related with the Weak Blaschke conjecture (C. T. Yang), that all Blaschke manifolds have right volumes. The author proves: Theorem. Suppose \(M^{2n}\) is a \(C_{2\pi}\)-manifold, homeomorphic to \(\mathbb{C} P^ n\). Then \(i(M) = i(\mathbb{C} P^ n,\text{can})\). The proof is based on equivariant …
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 703857
zbMATH Keywords: Blaschke manifold
zbMATH Keywords: closed geodesic
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: weak Blaschke conjecture
Wikidata QID: Q122937873
reviewed by: Vladimir Yu. Rovenskij
MaRDI profile type: Publication
cites work: Q4169382
cites work: Great circle fibrations of the three-sphere
cites work: Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics
cites work: Scattering of geodesic fields. II
cites work: Blaschke manifolds with taut geodesics
cites work: Q3726952
cites work: Blaschke manifolds of the projective plane type
cites work: On the volume of certain manifolds with closed geodesics
cites work: Q3198474
cites work: Q3825696
cites work: On the volume of manifolds all of whose geodesics are closed
cites work: The cut locus and conjugate locus of a riemannian manifold
cites work: Reduction of symplectic manifolds with symmetry
cites work: Odd-dimensional Wiedersehen manifolds are spheres
cites work: Q3948273
cites work: Smooth Great Circle Fibrations and an Application to the Topological Blaschke Conjecture
cites work: Isoperimetric constants and the first eigenvalue of a compact riemannian manifold
OpenAlex ID: W2079571839
description: scientific article; zbMATH DE number 703857