Abstract
Sekigawa proved in 1977 that a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the sense of I.M. Singer is always locally homogeneous. We deal here with the modification of the curvature homogeneity which is said to be “of type (1, 3)”. We give example of a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the modified sense but still not locally homogeneous.
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O. Kowalski’s research was supported by the grant MSM 0021620839 and by the grant from Grant Agency of Czech Republic GAČR no. P201/11/0356, A. Vanžurová’s research by the grant P201/11/0356 with the title: “Riemannian, pseudo-Riemannian and affine differential geometry”, by the project of specific university research of the Brno University of Technology, No. FAST-S-11-47, and by the grant MSM 6198959214 of the Czech Ministry of Education.
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Kowalski, O., Vanžurová, A. On a Generalization of Curvature Homogeneous Spaces. Results. Math. 63, 129–134 (2013). https://doi.org/10.1007/s00025-011-0177-y
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DOI: https://doi.org/10.1007/s00025-011-0177-y