## Hokkaido Mathematical Journal

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

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

Zdeněk DUŠEK and Oldřich KOWALSKI

#### 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

**Permanent link to this document**

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.)

**Keywords**

Einstein manifold 2-stein manifold Singer-Thorpe basis

#### 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