On a certain family of closed 13-dimensional Riemannian manifolds of positive curvature
One knows only very few diffeomorphism types of compact simply connected manifolds carrying a Riemannian metric of positive sectional curvature: the compact rank-one symmetric spaces, three other normal homogeneous spaces of dimensions 7, 7, 13, two infinite families of homogeneous and inhomogeneous 7-spaces and the flag manifolds over \(\mathbb{K} P^2\) for \(\mathbb{K} =\mathbb{C}\), \(\mathbb{H}\), \(\mathbb{O}\) with dimensions 6, 12, 24. In the present article, the author has extended this list by an infinite family of 13-dimensional manifolds carrying inhomogeneous metrics of positive curvature. These are orbit spaces of free circle actions on \(X := SU(5)/Sp(2)\) (where \(Sp(2) \subset SU(4) \subset SU(5)\)) which can be described as follows. For any quintuple of integers \(q = (q_1,...,q_5)\) put \(s = \sum_i q_i\) and consider the action \(\phi_q\) of \(S^1\) on \(SU(5)\) given by \[ \phi_q(z)A = \text{diag}(z^{q_1},...,z^{q_5}){\cdot}A{\cdot}\text{diag}(1,1,1,1,z^s)^{-1} \] for \(z \in S^1 \subset \mathbb{C}\) and \(A \in SU(5)\) where \(\text{diag}(a_1,...,a_5)\) denotes the diagonal matrix with entries \(a_1,...,a_5\). Clearly, this action descends to an action on \(X\) which is free iff \(\text{diag}(z^{q_1},...,z^{q_5})\) is never conjugate (for no \(z \neq 1\)) to any matrix of type \(\text{diag}(a, \bar a, b, \bar b, z^s)\), i.e., iff \(q_{\pi 1} + q_{\pi 2}\) are relatively prime to \(q_{\pi 3} + q_{\pi 4}\) for any permutation \(\pi \in S_5\). In fact, the author uses the slightly different action \[ \psi_p(z)A = \text{diag}(z^{p_1},...,z^{p_5}){\cdot}A{\cdot}\text{diag}(z^t,z^t,z^t,z^t,1)^{-1} \] where \(t = {1\over 4} \sum_i p_i\), but clearly we have \(\psi_p = \phi_q\) for \(p_i = q_i-s\). (Strictly speaking, the author considers a 2-torus action on \(U(5)/Sp(2)\) instead of an \(S^1\)-action on \(X = SU(5)/Sp(2)\) but this makes no difference for the orbit spaces.) In terms of the \(p_i\), the condition for the action to be free is that \(p_{\pi 1} + p_{\pi 2} - q_{\pi 3} - q_{\pi 4}\) and \(p_{\pi 5}\) are relatively prime for all \(\pi \in S_5\). If all the \(q_i\) (or \(p_i\)) are equal, the space \(X/\phi_q\) is homogeneous; in fact, it is the 13-dimensional normal homogeneous space of positive curvature mentioned above. This gives already a qualitative proof that we will obtain many new spaces of positive curvature for approximately equal \(q_i\). But the author exhibits even a precise domain of \(p\)'s such that \(X/\psi_p\) has positive curvature. However, the metric on \(X\) is not the normal homogeneous one but, similar to the 7-dimensional case, it is induced from a normal homogeneous metric on \(G = SU(5)\) represented as \(G = (G {\times} K)/\text{diag}(K)\) for \(K = S(U(4){\times} U(1)) \subset G\). For positive \(p_i\), this \(p\)-domain is given by the inequalities \[ \begin{aligned} p_{\pi 1} + p_{\pi 2} + q_{\pi 3} &> p_{\pi 4} + p_{\pi 5},\\ p_{\pi 1} + p_{\pi 2} + q_{\pi 3} + p_{\pi 4} &> 3p_{\pi 5},\\ 3(p_{\pi 1} + p_{\pi 2}) &> q_{\pi 3} + p_{\pi 4} + p_{\pi 5}.\end{aligned} \] Further, the integral cohomology of \(X/\psi_p\) is computed, using Leray-Serre spectral sequences. It depends on \(p\) only for \(H^6\) and \(H^8\) which are finite of order \(|\sigma_1^3 - 4\sigma_1\sigma_2 + 8\sigma_3|\) where \(\sigma_i\) is the \(i\)-th elementary symmetric function of \(p_1,...,p_5\). Moreover, we have \(H^i = {\mathbb{Z}}\) for \(i = 0,2,4,9,11,13\) and \(H^i = 0\) for all other \(i\). It might be interesting to note that \(X = SU(5)/Sp(2) = SU(6)/Sp(3)\) is a compact symmetric space with Dynkin diagram \(A_2\). The only other such spaces are \(X = SU(3)/SO(3),\;SU(3),\;E_6/F_4\); their noncompact duals are \(C(\mathbb{K} P^2)/I(\mathbb{K} P^2)\) for \(\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\) where ``\(C\)'' denotes the collineations of these projective planes and ``\(I\)'' the isometries. The infinite families in dimension 7 and 13 correspond to orbit spaces of circle actions on \(X\) for the cases \(\mathbb{K} = \mathbb{C}\) and \(\mathbb{K} = \mathbb{H}\). These are the only cases where there is a lower dimensional subgroup of \(I(X)\) acting transitively on \(X\). There are no such free circle actions for the other two cases; in particular, R. Bock has recently shown that no positive dimensional subgroup of \(E_6\) can act freely on \(E_6/F_4\).
scientific article; zbMATH DE number 1023881
scientific article