Hokkaido Mathematical Journal

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

Abstract

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 ($\mathbb{V}$, ⟨,⟩, 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.

Article information

Source
Hokkaido Math. J., Volume 44, Number 3 (2015), 441-458.

Dates
First available in Project Euclid: 1 August 2016

https://projecteuclid.org/euclid.hokmj/1470053374

Digital Object Identifier
doi:10.14492/hokmj/1470053374

Mathematical Reviews number (MathSciNet)
MR3532119

Zentralblatt MATH identifier
1339.53048

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citation

DUŠEK, Zdeněk; KOWALSKI, Oldřich. Transformations between Singer-Thorpe bases in 4-dimensional Einstein manifolds. Hokkaido Math. J. 44 (2015), no. 3, 441--458. doi:10.14492/hokmj/1470053374. https://projecteuclid.org/euclid.hokmj/1470053374