Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Dušek, Zdeněk

doi:10.1007/s10587-015-0230-1

Czech Math J

2015

DOI: 10.1007/s10587-015-0230-1

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Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek

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Transformations between Singer-Thorpe bases in 4-dimensional Einstein manifolds

Dušek¹,

Kowalski²

2015

Hokkaido Math. J.

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It is well known that, at each point of a 4-dimensional Einstein Riemannian manifold (M, g), the tangent space admits at least one so-called Singer-Thorpe basis with respect to the curvature tensor R at p. K. Sekigawa put the question "how many" Singer-Thorpe bases exist for a fixed curvature tensor R. Here we work only with algebraic structures (V, , , R), where , is a positive scalar product and R is an algebraic curvature tensor (in the sense of P. Gilkey) which satisfies the Einstein property. We give a partial answer to the Sekigawa problem and we state a reasonable conjecture for the general case. Moreover, we solve completely a modified problem: how many there are orthonormal bases which are Singer-Thorpe bases simultaneously for a natural 5-dimensional family of Einstein curvature tensors R. The answer is given by what we call "the universal Singer-Thorpe group" and we show that it is a finite group with 2304 elements.

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Transformations between Singer-Thorpe bases in 4-dimensional Einstein manifolds

Dušek¹,

Kowalski²

2015

Hokkaido Math. J.

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Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Cited by 1 publication

References 10 publications

Transformations between Singer-Thorpe bases in 4-dimensional Einstein manifolds

Transformations between Singer-Thorpe bases in 4-dimensional Einstein manifolds

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