The weak Blaschke conjecture for \(\mathbb{C} P^ n\)
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 (co)homology of the loop space [the author, Sov. Math., Dokl. 32, 100-103 (1985); translation from Dokl. Akad. Nauk SSSR 283, 308-312 (1985; Zbl 0595.53048)] for model spaces \(\mathbb{H} P^ n\) and \(\text{C}aP^ 2\)) and the idea (C. T. Yang for Blaschke manifolds modelled after \(\mathbb{C} P^ n\)), that there exist two integer classes \(p\) and \(q\) of order \(n + 1\) with the property \(p + q = \omega\), where \(\omega \in H^ 2(CM,\mathbb{Z})\) is the Euler class of \(S^ 1\)-fibration from the unit tangent bundle \(UM\) onto \((2n - 2)\)-dimensional space \(CM\) of oriented geodesics of \(M\), and there is the formula \(i(M) = 2(\omega^{2n-1},[M])\).
scientific article; zbMATH DE number 703857
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