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Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

Published online by Cambridge University Press:  17 April 2009

Y. Nikolayevsky
Y. Nikolayevsky
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Vic 3086, Australia e-mail: [email protected]
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A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 301 - 319
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Besse, A., Einstein manifolds, Ergebnisse der Mathematik and ihrer Grenzgebiete (3) 10 (Springer-Verlag, Berlin, 1987).CrossRefGoogle Scholar
[2]Bueken, P., ‘Three-dimensional Riemannian manifolds with constant principal Ricci curvatures ρ1 = ρ2 ≠ ρ3’, J. Math. Phys. 37 (1996), 4062–4075.CrossRefGoogle Scholar
[3]DeTurck, D. and Goldschmidt, H., ‘Metrics with prescribed Ricci curvature of constant rank. I. The integrable case’, Adv. Math. 145 (1999), 1–97.CrossRefGoogle Scholar
[4]DeTurck, D., ‘Existence of metrics with prescribed Ricci curvature: local theory’, Invent. Math. 65 (1981/1982), 179–207.CrossRefGoogle Scholar
[5]Gilkey, P., Leahy, J. and Sadofsky, H., ‘Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues’, Indiana Univ. Math. J. 48 (1999), 615–634.CrossRefGoogle Scholar
[6]Gilkey, P., ‘Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues II’, in Differntial Geometry and Applications (Masaryk Univ, Brno, 1999), pp. 73–87.Google Scholar
[7]Gilkey, P. and Semmelmann, U., ‘Spinors, self-duality, and IP algebraic curvature tensors’, (ESI preprint 616, 1998).Google Scholar
[8]Gilkey, P., Geometric properties of natural operators defined by the Riemann curvature tensor (World Scientific Publishing Co. Inc., River Edge, NJ, 2001).CrossRefGoogle Scholar
[9]Gilkey, P. and Ivanova, R., ‘The geometry of the skew-symmetric curvature operator in the complex setting’, in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math. 288 (American Mathematical Society, Providence, R.I., 2001), pp. 325–333.CrossRefGoogle Scholar
[10]Gilkey, P. and Ivanova, R., ‘Complex IP pseudo-Riemannian algebraic curvature tensors’, in PDEs, submanifolds and affine differential geometry (Warsaw, 2000), Banach Center Publ. 57 (Polish Acad. Sci., Warsaw, 2002), pp. 195–202.CrossRefGoogle Scholar
[11]Gilkey, P. and Zhang, T., ‘Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form’, Houston J. Math. 28 (2002), 311–328.Google Scholar
[12]Ivanov, S. and Petrova, I., ‘Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues’, Geom. Dedicata 70 (1998), 269–282.CrossRefGoogle Scholar
[13]Kowalski, O., Generalized symmetric spaces, Lecture Notes in Mathematics 805 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).CrossRefGoogle Scholar
[14]Kowalski, O., ‘A classification of Riemannian 3-manifolds with constant principal Ricci curvatures ρ1 = ρ2 ≠ ρ3’, Nagoya Math. J. 132 (1993), 1–36.CrossRefGoogle Scholar
[15]Kowalski, O. and Sekizawa, M., ‘Local isometry classes of Riemannian 3-manifolds with constant Ricci eigenvalues ρ1 = ρ2 ≠ ρ3 > 0’, Arch. Math. (Brno) 32 (1996), 137–145.Google Scholar
[16]Kowalski, O. and Nikčević, S., ‘On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds’, Geom. Dedicata 62 (1996), 65–72.CrossRefGoogle Scholar
[17]Milnor, J., ‘Curvatures of left invariant metrics on Lie groups’, Adv. In Math. 21 (1976), 293–329.CrossRefGoogle Scholar
[18]Nagata, M., ‘A remark on the unique factorization theorem’, J. Math. Soc. Japan, 9 (1957), 143–145.Google Scholar