1. Darmois, G.: Mémorial de Sciences Mathématiques, Fascicule XXV, “Les equations de la gravitation einsteinienne” (1927)
2. Readers unfamiliar with these terms can find a very clear and detailed explanation in the classic text by Eisenhart. See L. P. Eisenhart An Introduction to Differential Geometry. Princeton University Press (1947)
3. Lichnerowicz, A.: Théories Relativistes de la Gravitation et de l’Electromagnétisme Masson, Paris (1955)
4. These have been in use in spacetime for a century. For a convenient resourse see, for example, Misner, C., Thorne, K., Wheeler, J.: Gravitation. W.H. Freeman and Company. Section 21.13 (1973)
5. This has to be considered the most trivial case of a junction problem possible and many specific examples of this type of junction can be found. See, for example, A. Lightman, W. Press, R. Price and S. Teukolsky Problem Book in Relativity and Gravitation. Princeton University Press, Problem 9.29. They consider the special case $$g_{i j}=a(n)^2 \gamma _{i j}(x^{k})$$ g i j = a ( n ) 2 γ i j ( x k ) (1975)