Does the shape of the Universe refer to the curvature of spacetime in 5-dimensional space? [duplicate]

Question

When we say that the universe is flat, this means that the sum of angles of any triangle will always be 180°.

I believe that "flat" refers to the bending of spacetime in a 5-dimensional space (or a forth space dimension). This would however contradict the fact of the su of angles, as for warped space, the sum is smaller than 180°. Or does the geometry to the universe as a whole and "small" fluctuations like galaxies (which bend space) are ignored?

So: Is it correct to describe the shape of the universe in relation to a 5-dimensional surrounding? Like you can see in this picture: (Image Credit ESA)

In this picture, one space dimension is omitted (so the spheres actually should be circles), so our 3-dimensional Universe/Brane "floats" in a higher-dimensional hyperspace or bulk. In This image, space is locally warped by mass (or energy, but this is not shown here), but space as a whole is flat.

Is it correct to imagine the flat shape of the universe like this (theoretically with one space dimension more)?

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Update This question was closed as a duplicate of the following: Question 1, Question 2, Question 3.

While they are somewhat related to my question, they still ask for a different question, namely the following:

Question 1: Is your 3-dimensional universe part of the surface of a 4-dimensional sphere [like in the ant-sphere analogy] that we cannot perceive or access?

Question 2: The correctness of the bend-sheet-analogy for GTR

Question 3: Could the universe be bend over a forth dimension to form a 4-dimensional shphere?

The essence of my question was: When we refer to the shape of the universe (being flat for example), do we mean the same curvature as in GTR?

Answer 1

No, your belief is not correct. We do not, at least in General Relativity (GR), embed our spacetime in a higher dimensional space (or like you said in 4 dimensions of space).

Although I agree that it is possible to imagine many curved surfaces as embedded in a higher dimension, it is not how we do GR. In fact, the picture you have is one of the most misleading to interpret the mathematics of GR.

So what is happening is that you are restricted to your 4 dimensional manifold and you do not know what is outside it, like an ant restricted on a sphere would just imagine it as a 2 dimensional space and would not know that it is embedded in a 3 dimensional space.

Now to deal with such problems, Gauss had found the correct mathematical machinery which was refined by Reimann. In fact, the result you state about the sums of angle of a triangle on a curved surface is derived without embedding the surface in a higher dimension. We manage to figure out whether the space is curved or not by staying in that space and not seeing it from outside (by embedding).

The mathematics begins with the Gauss-Bonnet theorem and then leading to Reimanian geometry. What we calculate is the intrinsic curvature. For example: imagine a cylinder, you might see it as curved but it isn't a curved surface. It has zero intrinsic curvature. To get to that purely mathematically you need to show that Reimann curvature tensor vanishes but you can also see that intuitively. On the other hand a sphere is curved.

The cylinder has an extrinsic curvature (which can be calculated by embedding it) but no intrinsic curvature whereas a sphere has an intrinsic curvature.

GR is formulated in the language of intrinsic curvature. There's certainly nothing wrong with studying, say, a 2-sphere embedded in 3-dimensional space. But it's not necessary and requiring that such a higher-dimensional space even exists is an undue constraint. It's quite marvelous to realize that a 2-sphere can simply exist in nothing more than 2-dimensions: the geometry is encoded on the surface.

Answer 2

There is no need for a higher-dimensional space in which to embed the spatial manifold. The Riemann curvature is a measure of the intrinsic curvature of the surface -- it is independent of and does not require any embedding.

The Riemann tensor is the fundamental quantity that describes the intrinsic curvature of surfaces. A nice way to visualize how it "measures" curvature intrinsically (without reference to an embedding space), is to examine how a single vector, $V^\mu$, ends up when it is parallel transported along two different curves, $C$ and $C'$. The following picture is from Nakahara 7.3:

Starting a $p$, parallel transport of $V^\mu(p)$ to $q$ a distance $\epsilon$ away along $C$ gives $V^\mu_C(q) = V_0^\mu - V_0^\kappa \Gamma^\mu_{\nu \kappa}(p)\epsilon^\nu$. Then, along from $q$ a distance $\delta$ to $r$ gives $$V^\mu_C(r) = V^\mu_C(q)-V^\kappa_C(q)\Gamma^\mu_{\nu \kappa}(q)\delta^\nu$$ which we can write as $$V^\mu_C(r) \simeq V_0^\mu - V_0^\kappa\Gamma^\mu_{\nu \kappa}(p)\epsilon^\nu - V_0^\kappa\Gamma^\mu_{\nu \kappa}(p)\delta^\nu - V_0^\kappa[\partial_\lambda \Gamma^\mu_{\nu \kappa}(p)-\Gamma^\rho_{\lambda \kappa}(p)\Gamma^\mu_{\nu \rho}(p)]\epsilon^\lambda\delta^\nu$$ where we've kept terms up to second-order in $\epsilon$ and $\delta$.

You can do the same exercise along the other curve, $C'$. Then, when you take the difference of the vectors at the point $r$ you get $$V^\mu_{C'}(r) - V^\mu_C(r) = V_0^\kappa[\partial_\lambda \Gamma^\mu_{\nu \kappa}(p) - \partial_\nu \Gamma^\mu_{\lambda \kappa}(p) - \Gamma^\rho_{\lambda \kappa}(p)\Gamma^\mu_{\nu \rho}(p) + \Gamma^\rho_{\nu \kappa}(p)\Gamma^\mu_{\lambda \rho}(p)]\epsilon^\lambda \delta^\nu = V_0^\kappa R^\mu_{\kappa \lambda \nu}\epsilon^\lambda \delta^\nu,$$ where $R^\mu_{\kappa \lambda \nu}$ is the Riemann tensor. And so we can think of the Riemann curvature as arising from the fact that the orientation of a vector undergoing parallel transport depends on the path taken on curved surfaces. Importantly, there is no reference to any embedding space.

Of course, it is often helpful to visualize spaces with positive spatial curvature as spheres existing in a higher-dimensional space, but that's because we humans are used to seeing things this way. Part of the fun of differential geometry is learning to do away with these perceptual habits and understanding surfaces in terms of their intrinsic geometries.