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The idea of an oscillating Universe has remained a topic of interest evenafter the discovery of dark energy. This paper confirms this idea by means ofanother well-established theory in general relativity, the embedding of curvedspacetimes in higher-dimensional flat spacetimes: an $n$-dimensional Riemannianspace is said to be of embedding class $m$ if $m+n$ is the lowest dimension ofthe flat space in which the given space can be embedded; here$m=\frac{1}{2}n(n+1)$. So a four-dimensional Riemannian space is of class twoand can therefore be embedded in a six-dimensional flat space. A line elementof class two can be reduced to a line element of class one by a suitablecoordinate transformation. The extra dimension can be either spacelike ortimelike, leading to accelerating and decelerating expansions, respectively.Accordingly, it is proposed in this paper that the free parameter occurring inthe transformation be a periodic function of time. The result is a mathematicalmodel that can be interpreted as a periodic change in the signature of theembedding space. This signature change may be the best model for an oscillatingUniverse and complements various models proposed in the literature.

Periodic signature change in spacetimes of embedding class one