The global geometry of Riemannian manifolds with commuting curvature operators

Miguel Brozos-Vazquez and Peter B. Gilkey

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Submission date: 07. Sep. 2006
Pages: 10
published in: Journal of fixed point theory and applications, 1 (2007) 1, p. 87-96 
DOI number (of the published article): 10.1007/s11784-006-0001-6
Bibtex
MSC-Numbers: 58B20
Keywords and phrases: algebraic curvature tensor, geodesic completeness, scalar curvature blowup, skew Tsankov manifold
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Abstract:
We exhibit manifolds whose Riemann curvature operators commute, i.e. which satisfy the identity R(x,y)R(z,w)=R(z,w)R(x,y) for all x,y,z,w. We work in both the Riemannian and in the higher signature settings. These manifolds have global geometric properties which are quite different in the higher signature setting than in the Riemannian setting. Questions of geodesic completeness and the behaviour of the exponential map are investigated as are other analytic properties.