Acta Mathematica

Riemann spaces of class one and their characterization

T. Y. Thomas

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Acta Math. Volume 67 (1936), 169-211.

First available in Project Euclid: 31 January 2017

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1936 © Almqvist & Wiksells Boktryckeri-A.-B.


Thomas, T. Y. Riemann spaces of class one and their characterization. Acta Math. 67 (1936), 169--211. doi:10.1007/BF02401741.

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  • See. T. Y. Thomas, Un corollaire du théorème de Riquier, Bull. des. Sci. Math. 59, 1935, p. 134.
  • Trans. Am. Math. Soc., 38, 1935, p. 501.
  • On a class of existence theorems in differential geometry, Bull. Am. Math. Soc. 35, 1934, p. 721.
  • Trans. loc. cit. Trans. Am. Math. Soc., 38, 1935, p. 501. and T. Y. Thomas, On the metric representations of affinely connected spaces, Bull. Am. Math. Soc., 42, 1936, p. 77. These papers contain a proof of the non-existence of an algebraic characterization of the metric spaces in the class of all complex affinely connected spaces.
  • Beitrage zum Klassenproblem der quadratischen Differentialformen, Math. Annalen, 110, 1935, p. 522.
  • The conditions defining the algebraic characterization are considered to be known if they are difinitely obtainble by the recognized procedures of algebraic elimination.
  • Cf. Duschek-Mayer, Lehrbuch der Differentialgeometrie, II, Riemannsche Geometrie, Teubner, 1930. Eisenhart, Riemannian Geometry, Princeton University Press, 1926. Levi-Civita, The Absolute Differential Calculus, Blackie and Son, 1929.
  • See, for example, T. Y. Thomas, The Differential Invariants of Generalized Spaces, Cambridge University Press, 1934, p. 44, Eq. (13.8). Attention is called to the fact that the above components of the curvature tensor are the negatives of those used by some writers.
  • See, for example, T. Y. Thomas, Systems of total differential equations defined over simply connected domains, Annals of Math., 35, 1934, p. 730.
  • For example, if we make a linear transformation of the coordinates xα so that at the initial point we have gαβ=δ ${}_{β}^{α}$ with respect to the new coordinate system the equations (3.3) will be satisfied by taking σ1=⋯=σn=o,σn+1=I and ∂ ${}_{y}^{i}$ /∂xa=δ ${}_{a}^{i}$ . Transforming back to the orginal coordinates the transformed values of the quantities ∂yi/∂xα and the above values of the scarlars σi will satisfy the system (3.3) as required.
  • This theorem has been proved by Killing, Nicht Euklidische Raumformen, Leipzig (1885), p. 237 and by later writers; see, for example, L. P. Eisenhart, Riemannian Geometry, Princeton, (1926), p. 201. The proof which we have given of this theorem although somewhat more lengthy than the proofs of the above authors has the advantage of greater formal simplicity.
  • See B. L. van der Waerden, Moderne Algebra, II, Berlin, Springer, 1931, p. 14.
  • See M. Bocher, Introduction to Higher Algebra, MacMillan, 1929, p. 33.
  • Although we have previously defined the algebraic characterization in terms of conditions involving a set of polynomials in the structural functions gαβ of the Riemann space and their derivatives there is no objection to stating these conditions in terms of polynomials in the components of the curvature tensor B since we can immediately pass from these latter conditions to the former. Analogous remarks apply to the algebraic characterizations given by the following theorems.
  • It is here to be noted that we have avoided the use of the equations (9.6) and (9.7) which are applicable only when the space R2 is of definite type τ over H3. The method of § 9 involving the use of the above equations (9.6) and (9.7) enables ns however to avoid the transformation of coordinates in the Lemma of § 9 in the rigorous derivation of the continuity and differentiability properties of the solutions bαβ of the Gauss equations for spaces R ${}_{2}^{*}$ of type three; also the above equations are clearly necessary in the derivation of the conditions (10.2).