Abstract
Let R be an algebraic curvature tensor for a non-degenerate inner product of signature (p,q) where q≥5. If π is a spacelike 2 plane, let R(π) be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(ċ) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. We also classify the Ivanov–Petrova algebraic curvature tensors of rank 2; these are the algebraic curvature tensors of constant rank 2 such that the complex Jordan normal form of R(ċ) is constant.
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REFERENCES
N. BLAZIC, N. BOKAN and P. GILKEY, A Note on Osserman Lorentzian manifolds, Bull. London Math. Soc. 29 (1997), 227–230.
N. BLAZIC, N. BOKAN, P. GILKEY and Z. RAKIC, Pseudo-Riemannian Osserman manifolds, J. Balkan Society of Geometers 2 (1997), 1–12.
Q.-S. CHI, A curvature characterization of certain locally rank-one symmetric spaces, J. Differential Geom. 28 (1988), 187–202.
E. GARC A-RIO, D. KUPELI and M. VAZQUEZ-ABAL, On a problem of Osserman in Lorentzian geometry, Differential Geom. Appl. 7 (1997), 85–100.
E. GARC A-RIO, M. E. VAZQUEZ-ABAL and R. VAZQUEZ-LORENZO, Nonsymmetric Osserman pseudo-Riemannian manifolds, Proc. Amer. Math. Soc. 126 (1998), 2763–2769.
P. GILKEY, Geometric properties of the curvature operator, in: Geometry and Topology of Submanifolds, Vol. X, edited by W.H. Chen, A.-M. Li, U. Simon, L. Verstraelen, C.P. Wang, and M. Wiehe, World Scientific, Singapore, 2000, 62–70.
P. GILKEY, Manifolds whose curvature operator has constant eigenvalues at the basepoint, J. Geom. Anal. 4 (1994), 155–158.
P. GILKEY, Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues II, in: Differential geometry and applications, ed. by Kolar, Kowalski, Krupka, and Slovák, Publ Massaryk University Brno Czech Republic ISBN 80–210–2097–0, (1999), 73–87.
P. GILKEY, Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor, World Scientific (to appear Winter 2002).
P. GILKEY, J. V. Leahy and H. SADOFSKY, Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues, Indiana Univ. Math. J. 48 (1999), 615–634.
P. GILKEY and U. SEMMELMAN, Spinors, self-duality, and IP algebraic curvature tensors of rank 4, in: Proceedings of the Symposium in Contemporary Mathematics in honor of 125 years of faculty of Math at Univ. Belgrade, edited by Neda Bokan, 2000, 1–12.
S. IVANOV and I. PETROVA, Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues, Geom. Dedicata 70 (1998), 269–282.
O. KOWALSKI, M. SEKIZAWA, and Z. VLASEK, Can tangent sphere bundles over Riemannian manifolds have strictly positive sectional curvature? (preprint).
R. OSSERMAN, Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731–756.
T. ZHANG, Manifolds with indefinite metrics whose skew-symmetric curvature operator has constant eigenvalues, Ph.D. thesis, University of Oregon, 2000.
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Gilkey, P., Zhang, T. Algebraic curvature tensors whose skew-symmetric curvature operator has constant rank 2. Periodica Mathematica Hungarica 44, 7–26 (2002). https://doi.org/10.1023/A:1015221317388
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DOI: https://doi.org/10.1023/A:1015221317388