The even dimensional pinching problem and SU(3)/T
Let M be a compact and connected Riemannian manifold. This manifold M is called \(\delta\)-pinched if its sectional curvature \(\sigma\) (\(\lambda)\) satisfies the inequalities \(0<\delta \leq \sigma (\lambda)\leq 1\) for every plane \(\lambda \in T_ p(M)\) and for every \(p\in M\). The simply connected homogeneous manifolds which admit (homogeneous) metrics with positive sectional curvature have been classified by M. Berger, N. Wallach and P. Berard-Bergery. Aside from the rank 1 symmetric spaces, there are two normal homogeneous spaces \(M^ 7=Sp(2)/SU(2),\) and \(M^{13}=SU(5)/(Sp(2)\times S^ 1),\) three even-dimensional manifolds, \(M^ 6=SU(3)/T,\quad M^{12}=Sp(3)/SU(2)\times SU(2)\times SU(2),\) \(M^{24}=F_ 4/Spin(8)\) and an infinite family of seven manifolds \(SU(3)/S^ 1_{k,\ell}\). Lower bounds have been computed for the above- mentioned odd-dimensional examples, which are 1/37, 16/29 \(\cdot 37\) and 16/29 \(\cdot 37\) for the manifolds Sp(2)/SU(2), \(SU(5)/(Sp(2)\times S^ 1)\) and \(SU(3)/S^ 1_{k,\ell}\) respectively. The main result of this paper can be stated as follows. ``Any homogeneous \(\delta\)-pinched metric on SU(3)/T satisfies \(\delta\leq 4/21''\).
scientific article; zbMATH DE number 4096308
scientific article