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The decomposition of Riemann spaces in the large. (English) Zbl 0021.26001


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[1] A. Duschek–W. Mayer, Lehrbuch der Differentialgeometrie Bd. II: Riemannsche Geometrie, Teubner, 1930, p. 147. Also L. P. Eisenhart, Trans. Amer. Soc., vol. 25, 1923, p. 297.
[2] The summation is employed in this paper for the range 1, ...,n of the indices. It is moreover to be assumed that all indices have the range 1, ...,n unless the contrary is especially stated or unless it appears to be clear from the text that the indices have a different range without special mention.
[3] The statement thatR is of classC m means that the components of the fundamental metric tensor are of classC m as functions of the coordinates of the allowable coordinatc neighborhoods coveringR. IfR is of classC m this implies that the relationships between the coordinates of allowable coordinate neighborhoods are of classC m+1. In particular ifR is of classC orC {\(\omega\)} the coordinate relationships are also of classC {\(\omega\)} orC respectively. The convenient notationC {\(\omega\)} for an analytic Riemann space was introduced by Veblen and Whitehead: Foundations of Differential Geometry, Cambridge Tract,1932, Nr. 29.
[4] H. Hopf and W. Rinow, Über den Begriff der vollständigen differential-geometrischen Fläche. Commentarii Mathematici Helvetici, vol. 3, 1931, p. 209. · Zbl 0002.35004
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