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Algebraic curvature tensors of Einstein and weakly Einstein model spaces. (English) Zbl 1416.15018

Summary: This research investigates the restrictions on the symmetric bilinear form \(\varphi\) with associated algebraic curvature tensor \(R=R_\varphi\) in Einstein and weakly Einstein model spaces. We show that if a model space is Einstein and has a positive definite inner product, then: if the scalar curvature is non-negative, the model space has constant sectional curvature, and if the scalar curvature is negative, the matrix associated to the symmetric bilinear form can have at most two eigenvalues. We also show that, given \(R=R_\varphi\), a model space is weakly Einstein if and only if \(R_{\varphi^2}\) has constant sectional curvature.

MSC:

15A63 Quadratic and bilinear forms, inner products
15A69 Multilinear algebra, tensor calculus
15A18 Eigenvalues, singular values, and eigenvectors
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
83C99 General relativity
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References:

[1] T. Arias-Marco, O. Kowalski, Classification of 4-dimensional homogeneous weakly Einstein manifolds, Czechoslovak Math. J., 65 (2015), 21-59. · Zbl 1363.53032
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[4] P. Gilkey, Geometric Properties of Natural Operators Defined by the Riemannian Curvature Tensor, World Scientific, 2001. · Zbl 1007.53001
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