On a certain family of closed 13-dimensional Riemannian manifolds of positive curvature
type: Item
instance of: scholarly article
title: On a certain family of closed 13-dimensional Riemannian manifolds of positive curvature
zbMATH Open document ID: 0874.53034
DOI: 10.1007/BF02106732
published in: Siberian Mathematical Journal
publication date: 1997-09-18T00:00:00Z
review text: 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_{\…
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 55T10
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1023881
zbMATH Keywords: normal homogeneous metrics
zbMATH Keywords: free circle actions
zbMATH Keywords: biquotients
zbMATH Keywords: unitary matrices
zbMATH Keywords: integral lattice
zbMATH Keywords: spectral sequences
Wikidata QID: Q115392764
author: Yaroslav V. Bazaikin
reviewed by: J.-H. Eschenburg
MaRDI profile type: Publication
cites work: Q3283409
cites work: Compact homogeneous Riemannian manifolds with strictly positive curvature
cites work: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
cites work: Les variétés riemanniennes homogenes simplement connexes de dimension impaire à courbure strictement positive
cites work: Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature
cites work: New examples of manifolds with strictly positive curvature
cites work: Inhomogeneous spaces of positive curvature
cites work: The fundamental equations of a submersion
cites work: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts
full work available at URL: https://doi.org/10.1007/bf02106732
OpenAlex ID: W1964721974
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description: scientific article; zbMATH DE number 1023881