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Homogeneous metrics with the same curvature. (English) Zbl 0762.53034

Two homogeneous Riemannian manifolds are said to have the same curvature if there exists a linear isometry between tangent spaces (for some points in the respective manifolds) which preserves the Riemann curvature tensor. The question here is that if two homogeneous Riemannian manifolds have the same curvature, are they locally isometric? In the case of dimension two the answer is positive. The author proves that in general the answer is negative, already in dimension three. This is done by showing that some Riemannian invariants are different.

MSC:

53C30 Differential geometry of homogeneous manifolds
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