Geometries involving affine connections and general linear connections. An extension of the recent Einstein-Mayer geometry
This paper deals with a general linear connection in the sense of R. König, in which a space of \(m\) dimensions is attached to each point of a general manifold of \(n\) dimensions. In such a general manifold of \(n\) dimensions are assumed a symmetric linear connection \(\Gamma_{jk}^i\), and a general linear connection \(L_{\beta a}^\alpha\), both functions of the coordinates \(x^1, x^2, \ldots x^n\), and where the Greek indices run from \(1\) to \(m\), the Latin from \(1\) to \(n\). The definition of \(L\) is in accordance with suggestions by \textit{J. H. C. Whitehead} [Trans. Am. Math. Soc. 33, 191--209 (1931; Zbl 0001.16703; JFM 57.0908.02)]. Then composite tensors are studied, defined as tensors which may have both Greek and Latin indices. With the aid of normal representations, normal tensors are constructed which lead to a general reduction theorem. A particular case is that of Einstein and Mayer in which \(n=5\), \(m=4\) and the \(\Gamma_{jk}^i\) are the Riemann-Christoffel symbols. The \(L_{\beta a}^\alpha\) can then be computed. Beside the general reduction theorem there exists also a reduction theorem for tensor differential invariants with only Latin indices.
scientific article; zbMATH DE number 3010469