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Reference16 articles.
1. See, for example,C. Fronsdal:Nuovo Cimento,13, 987 (1959); D. W. Joseph:Phys. Rev., 126, 319 (1962). See also several papers of the symposium on particle symmetries and the embedding problem, held at the Southwest Center for Advanced Studies (Dallas, Tex., 1964), published inRev. Mod. Phys., 37, 201 (1965).
2. In general, we say that an m-dimensional Riemannian metricg^ is of classp if the minimal flat embedding space of gμν is of dimensionsN =m + p, wherem ⩽N ⩽ ⩽1/2 m(m + 1); local embedding is here alluded, to be sure; cf.L. P. Eisenhart:Riemannian Geometry (Princeton, N. J., 1926), p. 187.
3. A. Friedman:Journ. Math. Mech.,10, 625 (1961).
4. It is not known, at present, whether, for a given indefinite Riemannian manifold, a global isometric embedding into some flat space with suitable signature is always possible. For positive-definite Riemannian metrics a global isometric flat embedding is always possible;J. Nash:Ann. Math.,63, 20 (1956).
5. Compare, for instance,E. Kasner:Am. Journ. Math.,43, 130 (1921), with C. Fronsdal:Phys. Rev., 116, 778 (1959), where the Schwarzschild line element is embedded in six dimensions, locally and globally, respectively, Kasner’s local embedding embraces the co-ordinate patch corresponding to the whole external Schwarzschild solution, although not the complete analytic extension of the Schwarzschild manifold, as Fronsdal does.
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