Wikipedia says that inflation is the exponential expansion of space in the early universe.I'm trying to have a physical picture of this.Given that I can't visualize 3+1 pseudoriemannian manifolds,I'm trying to understand the situation for 2+1 pseudoriemannian manifolds. Can we embed 2+1 space-time of GR in a 4 Dimensional flat Euclidean space ?

  • $\begingroup$ Related: physics.stackexchange.com/q/8932/2451. Note that the word Euclidean space has different meanings in mathematics and physics, cf. my Phys.SE answer here. $\endgroup$
    – Qmechanic
    Jan 27, 2015 at 9:33
  • $\begingroup$ Whether or not one can embed a manifold into a higher dimensional space is more subtle in physics than in mathematics since matter localised to the submanifold can affect the bulk which in turns affects whether or not an embedding is possible. $\endgroup$
    – JamalS
    Jan 27, 2015 at 9:35
  • $\begingroup$ I won't venture an answer because this is just a vague memory, but I'm sure I have read that a de Sitter space cannot be embedded in a n+1 Euclidean space without being self intersecting. It can be embedded in an n+1 Minkowski space. During inflation the universe was approximately de Sitter, so the above may be relevant. $\endgroup$ Jan 27, 2015 at 11:44
  • $\begingroup$ To reopen this question (v1) consider 1. clarifying your definition of Euclidean space, and 2. harmonize title question and question in main body (3D vs. 4D). $\endgroup$
    – Qmechanic
    Apr 29, 2016 at 19:47
  • $\begingroup$ Pullback of metric by a smooth immersion preserves signature (a pretty simple exercise, use the definition of the differential), therefore I'm pretty sure a Riemannian metric (e.g. Euclidean) can't induce a lorentzian metric on a submanifold. $\endgroup$
    – zzz
    Apr 29, 2016 at 23:00

1 Answer 1


I don't see why not. In fact one nice way to generate certain symmetric spacetimes is to embed them in higher dimensional space then constrain your variables to the manifold and massage the induced metric into something palatable. So I'm going to do you one better: I'm going to embed a 1+1 GR spacetime in 3D Euclidean space. The best example I can come up with off the top of my head is de Sitter space, which actually works as a good expanding universe model (which I will hedge by pointing out that this only matches the FRW metric in certain limits). de Sitter space is basically the Minkowski version of a sphere, where we fix the Minkowski distance squared of any point in $dS^N$ from the origin of $\mathbb{R}^{N,1}$ to be a constant, then massage the induced metric. $dS^2$ is then a hyperboloid of one sheet.


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