1. It was conjectured bySchläfli:Ann. Math., (2)5, 170 (1871), and proved byCartan:Ann. Soc. Polon. Math.,6, 1 (1927), that aV n is embeddable locally in anS m , wherem=1/2n(n+1); seee.g. L. P. Eisenhart:Riemannian Geometry (Princeton, 1926), p. 188. When we say that a space is embeddable in a higher dimensional space we mean that it may be embedded locally. Singular points of the embedding may occur, as well as singular lines, etc. The only restriction is that the manifold of singular points must be of dimension less than that of the embedded space. This is the terminology ofC. B. Allendoerfer:Duke Math. Journ.,3, 317 (1937).
2. E. Kasner:Am. Journ. Math.,43, 126 (1921).
3. E. Kasner:Am. Journ. Math.,43, 130 (1921).
4. The common cosmological solutions are all embeddable in anS 6, except for the Goedel solution. (Private communication fromR. J. Finkelstein). Having been unable to find a proof in the literature, we give a simple proof that the line-element $$ds^2 = (dx_1^2 + dx_2^2 + dx_3^2 )\psi ^2 (r,t)--dt^2 \varphi ^2 (r,t)$$ is embeddable in anS 6. In fact, takeZ i = ψx i , to obtain $$ds^2 = (dZ_1^2 + dZ_2^2 + dZ_3^2 + |\left( {\frac{{\partial r\psi }}{{\partial r}}} \right)^2 --\psi ^2 |dr^2 + 2\frac{{\partial r\psi }}{{\partial r}}\frac{{\partial r\psi }}{{\partial t}}drdt + |\left( {\frac{{\partial r\psi }}{{\partial t}}} \right)^2 --\varphi ^2 |dt^2 $$ Thus the problem has been reduced to that of embedding aV 2 in anS 3, which is always solvable (ref. (1)It was conjectured by ).
5. See for exampleE. L. Hill:Phys. Rev.,72, 143 (1947); andJ. A. Schouten:Rev. Mod. Phys.,21, 421 (1949).