Abstract
Let us recall the basic elementary data concerning Riemannian spaces. The material to be presented in this section is of auxiliary nature, and proofs of the relations appearing here can be found in known textbooks on Riemannian geometry.
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- 1.
Some authors use another definition of the Ricci tensor components, R jk = R ijk i, which differs in sign from definition (2.9) adopted here.
- 2.
However, a certain extension of the representation of orthogonal groups does exist. Thus, for instance, the group of linear transformations of a plane, defined by the matrices
$$\displaystyle \begin{aligned} M=\begin{Vmatrix} m_1 &-m_2\\ m_2 & m_1 \end{Vmatrix} , \end{aligned} $$has the representation M →{±S}, where S is defined in the following way:
$$\displaystyle \begin{aligned} S= \begin{Vmatrix} \sqrt{m_1+\mathrm{i} m_2}& 0 \\ 0 & 1/\sqrt{m_1+\mathrm{i} m_2} \end{Vmatrix} . \end{aligned} $$This representation passes into the spinor representation of the group of rotations of the plane for \(m_1=\cos \varphi \), \(m_2=\sin \varphi \) (see Chap. 4).
- 3.
The theory of parallel transport of spinors in Riemannian spaces may also be developed without this restriction (see [4]).
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Zhelnorovich, V.A. (2019). Spinor Fields in a Riemannian Space. In: Theory of Spinors and Its Application in Physics and Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-27836-6_2
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DOI: https://doi.org/10.1007/978-3-030-27836-6_2
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