An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures

Abstract

1. Introduction.The purpose of this note is to show that if T 1 is a closed, connected one-dimensional subgroup of SU(3) that has no nonzero fixed points, then SUtyjT 1 admits an SC/^-invariant Riemannian structure of strictly positive curvature.This result implies the statement of the title since there are an infinite number of distinct homotopy types among the spaces SUtyjT 1 with T 1 as above (see Lemma 3.3).To prove the above result we introduce what we call condition II, which generalizes condition III of [2] and [3].Although the spaces SUty/T 1 are the only new spaces satisfying condition II (see Theorem 5.1 below), it is worthwhile to introduce the notion since the 13-dimensional example, M 2 , of Berger [1] satisfies condition II in a nontrivial fashion.Hence there is a set of invariant metrics < , ) t on M 2 with -l<f<£ of strictly positive curvature (see §4) with ( , ) 0 the only normal one.A word should be said about the joint authorship of this paper.The main result of this note was found independently by the authors by slightly different methods.We have given the technique of the second author since it is conceptually simpler.Lemma 3.3 was pointed out to the second author by Professor Lashof of the University of Chicago and derived independently by the first author.The second author would like to also thank Professor M. P. do Carmo and the students of IMP A, Rio de Janeiro, for their help during his research on this problem.2. Condition II.Let G be a compact connected Lie group and let K be a closed subgroup of G.DEFINITION 2.1.(G, K) is said to satisfy condition II if there is an Ad

Locations

• Project Euclid (Cornell University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF