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**Positively curved manifolds with symmetry.**
*(English)*
Zbl 1104.53030

Only little is known about which simply connected compact manifolds admit Riemannian metrics with positive sectional curvature \(K > 0\). There are only few examples (in dimension \(> 24\) only the compact rank 1 symmetric spaces) and very few known general obstructions (Gromov’s Betti number theorem for manifolds with \(K \geq 0\)).

The paper under review studies compact simply connected manifolds with \(K > 0\) and large isometry groups. This subject came to life with M. Berger’s paper [Ann. Sc. Norm. Super. Pisa 15, 179–246 (1961; Zbl 0101.14201)], which was the starting point for the classification of the homogeneous spaces of positive sectional curvature by N. R. Wallach [Ann. Math. 96, 277–295 (1972; Zbl 0261.53033)] and L. Berard-Bergery [J. Math. Pur. Appl. 55, 47–68 (1976; Zbl 0289.53037)]. This classification revealed examples in dimensions 6, 7, 12, 13, and 24 (these were later supplemented by the inhomogeneous biquotients of J.-H. Eschenburg [Invent. Math. 66, 469–480 (1982; Zbl 0484.53031)] and Ya. V. Bazaikin [Sib. Math. J. 37, 1068–1085 (1996; Zbl 0874.53034)]).

The natural question is now which new examples occur when the symmetry assumptions are reduced. K. Grove suggested to do this reduction in various ways. Together with C. Searle, he classified manifolds with \(K > 0\) and maximal symmetry rank [J. Pure Appl. Algebra 91, 137–142 (1994; Zbl 0793.53040)] and fixed point homogeneous manifolds with \(K>0\) [J. Differ. Geom. 47, 530–559 (1997; Zbl 0929.53017)]. Among the several subsequent results in this spirit, we only mention L. Verdiani’s classification of even dimensional cohomogeneity one manifolds with \(K > 0\) [J. Differ. Geom. 68, 31–72 (2004; Zbl 1100.53033)]. All these classifications did not uncover new examples so far.

The present paper and its dizygotic twin [B. Wilking, Acta Math. 191, 259–297 (2003; Zbl 1062.53029)] make statements about such classifications in general. For example, it is not possible to find new manifolds with \(K > 0\) and cohomogeneity \(\leq k\) for fixed \(k\) in arbitrary high dimensions. More precisely, the main results derived in this paper are the following: If the dimension of the isometry group of a compact \(n\)-dimensional manifold \(M\) with \(K > 0\) is \(\geq 2n-6\) then \(M\) is tangentially homotopically equivalent to a rank one symmetric space or to a homogeneous space with \(K > 0\). If the rank of the isometry group is \(>3(\text{cohom}(M)+1)\) then \(M\) is homogeneous or tangentially homotopically equivalent to a rank one symmetric space. If \(\text{cohom}(M) = k > 1\) and \(\text{dim}(M) > 18(k+1)^2\) then \(M\) is tangentially homotopically equivalent to a rank one symmetric space.

At least as important as the results are the new tools developed in this paper. Like in its above mentioned twin a central role is played by the connectedness lemma of the author: A totally geodesic embedded submanifold \(N^{n-h} \to M^n\) is \((n-2h+1)\)-connected if \(M\) has positive sectional curvature. This and the several other tools have already found applications in a forthcoming paper by Grove, Wilking, and Ziller where the three authors classify the odd dimensional cohomogeneity one manifolds with \(K > 0\) up to a short list of \(7\)-dimensional promising candidates.

The paper under review studies compact simply connected manifolds with \(K > 0\) and large isometry groups. This subject came to life with M. Berger’s paper [Ann. Sc. Norm. Super. Pisa 15, 179–246 (1961; Zbl 0101.14201)], which was the starting point for the classification of the homogeneous spaces of positive sectional curvature by N. R. Wallach [Ann. Math. 96, 277–295 (1972; Zbl 0261.53033)] and L. Berard-Bergery [J. Math. Pur. Appl. 55, 47–68 (1976; Zbl 0289.53037)]. This classification revealed examples in dimensions 6, 7, 12, 13, and 24 (these were later supplemented by the inhomogeneous biquotients of J.-H. Eschenburg [Invent. Math. 66, 469–480 (1982; Zbl 0484.53031)] and Ya. V. Bazaikin [Sib. Math. J. 37, 1068–1085 (1996; Zbl 0874.53034)]).

The natural question is now which new examples occur when the symmetry assumptions are reduced. K. Grove suggested to do this reduction in various ways. Together with C. Searle, he classified manifolds with \(K > 0\) and maximal symmetry rank [J. Pure Appl. Algebra 91, 137–142 (1994; Zbl 0793.53040)] and fixed point homogeneous manifolds with \(K>0\) [J. Differ. Geom. 47, 530–559 (1997; Zbl 0929.53017)]. Among the several subsequent results in this spirit, we only mention L. Verdiani’s classification of even dimensional cohomogeneity one manifolds with \(K > 0\) [J. Differ. Geom. 68, 31–72 (2004; Zbl 1100.53033)]. All these classifications did not uncover new examples so far.

The present paper and its dizygotic twin [B. Wilking, Acta Math. 191, 259–297 (2003; Zbl 1062.53029)] make statements about such classifications in general. For example, it is not possible to find new manifolds with \(K > 0\) and cohomogeneity \(\leq k\) for fixed \(k\) in arbitrary high dimensions. More precisely, the main results derived in this paper are the following: If the dimension of the isometry group of a compact \(n\)-dimensional manifold \(M\) with \(K > 0\) is \(\geq 2n-6\) then \(M\) is tangentially homotopically equivalent to a rank one symmetric space or to a homogeneous space with \(K > 0\). If the rank of the isometry group is \(>3(\text{cohom}(M)+1)\) then \(M\) is homogeneous or tangentially homotopically equivalent to a rank one symmetric space. If \(\text{cohom}(M) = k > 1\) and \(\text{dim}(M) > 18(k+1)^2\) then \(M\) is tangentially homotopically equivalent to a rank one symmetric space.

At least as important as the results are the new tools developed in this paper. Like in its above mentioned twin a central role is played by the connectedness lemma of the author: A totally geodesic embedded submanifold \(N^{n-h} \to M^n\) is \((n-2h+1)\)-connected if \(M\) has positive sectional curvature. This and the several other tools have already found applications in a forthcoming paper by Grove, Wilking, and Ziller where the three authors classify the odd dimensional cohomogeneity one manifolds with \(K > 0\) up to a short list of \(7\)-dimensional promising candidates.

Reviewer: Thomas Püttmann (Bonn)

### MSC:

53C20 | Global Riemannian geometry, including pinching |