2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/34892We consider branes $N=I\times\so$, where $\so$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\hat N$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS_{(n+2)} bulk $\hat{\mc N}$. The joint branes $N\uu \hat N$ can thus be naturally embedded in $R^2\times \so$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N\uu\hat N$ forms a $C^\infty$-hypersurface in $R^2\times \so$. This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state.20 pages, a pdf version can also be retrieved from http://www.math.uni-heidelberg.de/studinfo/gerhardt/brane.pdf and bibtex data from http://www.math.uni-heidelberg.de/studinfo/gerhardt/bibtexcgbrane.html, v5: the results of http://arXiv.org/abs/gr-qc/0404112 have been applied to ARW branes, the introduction has been rewritten, this is the final version that will appear in Adv. Theor. Math. PhysicsGeneral Relativity and Quantum CosmologyHigh Energy Physics - TheoryDifferential Geometrytext