On One Family of 13-dimensional Closed Riemannian Positively Curved Manifolds

In the present paper we describe one family of closed Riemannian manifolds with positive sectional curvature. Now the list of known examples is not large (for instance, all known manifolds with dimension > 24 are diffeomorphic to compact rank one symmetric spaces) (we restrict ourselves only by pointing out simply connected manifolds) : 1) Berger described all normally homogeneous closed positively curved manifolds that are compact rank one symmetric spaces (i.e., the spheres S, the complex projective spaces CP, the quaternionic projective spaces HP, and the projective Cayley plane CaP ), and two exceptional spaces of form Sp(2)/SU(2) and SU(5)/Sp(2)× S with dimension 7 and 13, respectively (notice that the embedding SU(2) ⊂ Sp(2) is not standard) ([Be]); 2) Wallach had shown that all even-dimensional simply connected closed positively curved are diffeomorphic to normally homogeneous ones or the flag spaces over CP , HP , and CaP 2 (with dimension 6,12, and 24, respectively) ([W]); 3) Aloff and Wallach ([AW]) constructed infinite series of spacesNp,q of the form SU(3)/S where the subgroup S is a winding of a maximal torus of group SU(3) and, since that, is defined by a pair of relatively prime integer parameters p and q. If some conditions for these parameters p and q hold then these manifolds admit left-invariant homogeneous Riemannian metric with positive sectional curvature. Berard-Bergery ([BB]) had shown that the Aloff-Wallach spaces are all possible manifolds that admit homogeneous positively curved metric and do not admit normally homogeneous one, and Kreck and Stolz had found among them a pair of homeomorphic but nondiffeomorphic manifolds (N−56788,5227 and N−42652,61213 ; [KS]) ; 4) by using of the construction of Aloff and Wallach, Eschenburg had found an infinite series of seven-dimensional spaces with nonhomogeneous positively curved metrics ([E1]) and in the sequel had found an example of six-dimensional nonhomogeneous space with positively curved metric ([E2]).