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Geometries involving affine connections and general linear connections. An extension of the recent Einstein-Mayer geometry. (English) Zbl 0007.32801

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 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.

MSC:

53C05 Connections (general theory)
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References:

[1] Robert König, “ Jahresberichte der Deutschen Mathematiker Vereinigung {”, vol. 28, 1919, pp. 213–228.}
[2] L. Schlesinger, “ Mathematische Annalen {”, vol. 99, 1928, pp. 413–434.} · JFM 54.0766.04 · doi:10.1007/BF01459106
[3] J. A. Schouten, “ Proceedings Koninklijke Akad. v. Wetenschappen {”, Amsterdam, vol. 27, 1924, pp. 407–424.}
[4] J. H. C. Whitehead, “ Transactions of the Amer. Mathem. Society {”, vol. 33, 1931, pp. 191–209.} · doi:10.1090/S0002-9947-1931-1501584-3
[5] A. Einstein andW. Mayer, “ Sitzungsberichte der Preuss. Akad. {”, Dec. 1931, pp. 541–547. Ibid., April 14, 1932.}
[6] O. Veblen,Invariants of Quadratic Differential Forms, “ Cambridge Tract {”, 1927. See also,O. Veblen andT. Y. Thomas, “ Transactions of the Amer. Mathem. Society {”, vol. 25, 1923, pp. 561–608.}}
[7] L. P. Eisenhart,Non-Riemannian Geometry, “ Colloquium Lectures of the Amer. Mathem. Society {”, 1927.} · JFM 52.0721.02
[8] A. D. Michal, “ Transactions of the Amer. Mathem. Society {”, vol. 29, 1927, pp. 612–646; “ Proceedings of the National Academy {”, vol. 14, 1928, pp. 746–754; “ Bulletin of the Amer. Mathem. Society {”, vol. 36, 1930, pp. 541–546; “ Amer. Mathem. Monthly {”, vol. 37, 1930, pp. 529–533; “ Tôhoku Mathem. Journal {”, 1931; “ Proceedings of the National Academy {”, vol. 17, 1931, pp. 132–136.}}}}}}
[9] A. D. Michal andT. Y. Thomas, “ Annals of Mathematics {”, vol. 28, 1927, pp. 196–236. Ibid., vol. 28, 1927, pp. 631–688.} · JFM 53.0684.02
[10] E. Bortolotti, “ Annali di Mathem. {”, Serie IV, Tomo VIII, 1930–1931, pp. 54–101; “ Annals of Mathematics {”, vol. 32, 1931, pp. 361–377.}}
[11] T. Levi-Civita, “ Rendiconti del Circolo Mathematico di Palermo {”, vol. 42, 1917, pp. 173–205. This paper initiated the theory of parallel displacements in Riemannian geometry. See also his book,The Absolute Differential Calculus (Blackie & Son, 1927).} · doi:10.1007/BF03014898
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