Are there closed simply connected 2D manifolds that do not require a third dimension?

Question

In the context of cosmology, space is commonly described as potentially having a global curvature that can be positive, zero, or negative. A common way that textbooks describe positive curvature is by saying that one example of a 3D space with positive curvature would be the 'surface' of a 3-sphere. In the process of trying to understand this in more detail, I ran into the common question of whether a universe with global positive curvature, if we assumed the curvature was homogeneous on large scales, would need to imply the existence of a fourth spatial dimension for the universe to curve 'into' and from all the answers I have seen, it seems the answer is 'no' - that what we are measuring is the intrinsic curvature of space and that it does not imply that there would have to be extrinsic curvature into an additional dimension (but please correct me if I am wrong!).

What I don't quite understand is the following: if we consider a 2D universe in which inhabitants measured positive curvature that was consistent with what would be measured if their 2D universe lived on top of a 2-sphere, and also noticed that their universe was finite and had no boundaries (such that they could travel around their universe in a 'straight line' and come back to the starting point), could they not prove that there had to be a third dimension? In other words, are there 2D manifolds that would have these properties that do not require a third dimension and do not have extrinsic curvature? (all the examples of closed 2D manifolds I have found are defined on a 3D surface - sphere, torus, Klein bottle)

If the inhabitants of this 2D universe determined that there had to be a third dimension to make sense of the observed intrinsic curvature and boundary properties of their universe, why would it not also be the case that a 3D finite/closed universe with positive curvature require a fourth dimension? Could one explain a finite 3D universe with positive intrinsic curvature where you can travel in one direction and travel along a 'straight line' and come back to your starting point without a fourth dimension?

Answer 1

No. A $2$D manifold might fit perfectly well in a $3$D manifold if it exists. But that does not mean that it has to exist. There are a couple ways to approach this.

First "exists". Manifold are mathematical objects, not physical objects. There are physical objects whose properties match manifold pretty well, but the abstract idea of a manifold is a mathematical concept. Mathematicians invent abstract spaces all the time. What exists in those spaces is whatever they decide to include.

They might invent a number line that contains every point except $0$. This might be useful because for every number $x$ on this line, the number $1/x$ exists. They could have invented more points at $0$ and perhaps $\infty$. But they don't have to.

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Second, You are thinking that a sphere fits naturally into a $3$D manifold. But there is more than one way to embed it in a higher dimensional manifold.

A line fits naturally into a plane. A plane fits naturally into $3$D space. $3$D space fits naturally into $4$D space. And so on.

You can naturally fit a circle in a plane or in a torus. A torus is a $2$D manifold that is different from a plane.

There are higher dimensional analogs to a torus. You can embed a sphere into a $3$D torus. Or more complex manifolds. The existence of a sphere does not force the sphere to be embedded in all these manifolds at the same time.

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Even though higher dimensions don't have to exist for the reason you gave, one can ask if they do. Our spacetime matches the properties of an abstract $4$S manifold well. We often say spacetime is a $4$D manifold. Spacetime is curved by the presence of matter. You might well ask if spacetime could be embedded into a higher dimensional manifold.

If it can, this is not evidence that it is. You would need experimental evidence that it is.

String theories do treat spacetime with many more dimensions. The reasons for this are similar to the reason we work with a $4$D spacetime instead of a $3$D space and $1$D time. There are unexpected similarities between space and time that are best expressed with a $4$D spacetime.

Kaluza–Klein theory because when Kaluza noticed that a $5$D manifold could explain electromagnetism. Klein gave it a quantum interpretation by making the $5^{th}$ dimension toroidal with an extremely small diameter.

People have looked for ways to experimentally detect the existence of a $5$D dimension, so far without success.

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String theory builds on this idea.

General relativity is a theory of gravity. General relativity predicts matter will be crushed to a point at the center of a black hole. When sizes get very small, quantum mechanics becomes important. But general relativity has inconsistencies with quantum mechanics.

String theory is a way to combine the two theories in a consistent way. The fundamental particles are not points. They are tiny loops, or strings. The different vibrational patterns of the strings correspond to the different particles. You need more dimensions to provide enough vibrational patterns to explain all the properties of all the particles. Here is a Don Lincoln video on it Superstrings

Answer 2

Aside from @mmesser314’s nice answer let me mention some other results. From the classification of surfaces, we know that the only (non-empty) 2-dimensional compact boundaryless simply connected smooth manifold is $S^2$. Said differently:

if $\Sigma$ is a non-empty 2-dimensional, compact, boundaryless, simply-connected topological (resp. smooth) manifold, then $\Sigma$ is homeomorphic (resp. diffeomorphic) to $S^2$.

So, the 2-dimensional sphere is the only thing you have. Note that this is a purely (differential) topological result. This makes no reference whatsoever to curvature or embeddings or whatsoever.

A standard fact about the sphere $S^2$ is that with its usual unit round metric tensor, it has constant sectional curvature $+1$ (this is a fully intrinsic notion). In fact, a nice topological result (e.g this or this) tells us something more: there is NO connection on $S^2$ with zero curvature… or said differently, $S^2$ doesn’t admit flat connections, or even more plainly, there is no sense in which $S^2$ is intrinsically flat. Once again, these are purely intrinsic statements.

Next, suppose we have an embedding of $S^2$ into a flat manifold $M$ (i.e $M$ is a manifold equipped with a flat connection). Then Gauss’ equation combined with the above result tells us that the extrinsic curvature (i.e the shape tensor) has to be non-zero. Roughly speaking, Gauss’ equation tells us how the ambient (intrinsic) curvature is related to both the intrinsic and extrinsic curvature of a submanifold, (i.e it relates three quantities). So if one of them is zero (e.g the ambient curvature) and one of the them is non-zero (the intrinsic curvature of the submanifold) then the other has to be non-zero as well (the extrinsic curvature of the submanifold).

But anyway even before all of these more ‘advanced’ results, I think you need to first of all appreciate the basic definitions of smooth manifold, (semi-)Riemannian metric, Levi-Civita connection, shape-tensor etc so that you understand what exactly the intrinsic notions are (and you’ll see that the extrinsic notions, even though historically the first ones discovered/introduced, are only introduced very late on in modern treatments of geometry).

Answer 3

You wrote All the examples of closed 2D manifolds I have found are defined on a 3D surface.

From a mathematical perspective, there are several constructions of closed 2D manifolds, and closed manifolds of any dimension, that do not require more dimensions. Constructions of this type are taught in topology courses.

Perhaps one of the more familiar examples is a construction of the torus as a quotient space of the 2-dimensional Euclidean plane $\mathbb R^2$, by requiring that each point $(x,y)$ of $\mathbb R^2$ be regarded as equivalent to all points of the form $(x+m,y+n)$, using the same values of $x$ and $y$ but with arbitrary integer values of $m,n$. Escher drew many interesting images of toruses in this pattern; for example here is an example of such a torus inhabited by two creatures: a white goose, and a blue goose.

That torus construction is not a simply connected example as said in your title.

But there are other "quotient space" constructions which do produce, say, the 2-dimensional sphere. For example, you can take two separate 2-dimensional discs in the Euclidean plane, and then abstractly identify the boundary circle of one disc with the boundary circle of the other. What you get is topologically identical to a standard round $2$-sphere embedded in 3-dimensional space, subdivided into a northern hemisphere and a southern hemisphere.

Answer 4

There are two distinct aspects to the question you are asking about representing a closed simply-connected curved 2D surface like a sphere without requiring extra dimensions.

One is topological. The topology of a sphere is different to a plane, and there is no subset of the plane that can be mapped smoothly to a sphere. Topology has no awareness of a surface's curvature. The obstruction has nothing to do with curvature. It doesn't even really have anything to do with the number of dimensions. A Klein bottle cannot be embedded in flat 3D space without self-intersection, even though the Klein bottle is 2D and space is 3D.

The other aspect is to do with the metric definition of curvature. We are all intuitively familiar with the usual Euclidean definition of lengths and distances, but it is not the only way you can define it. A familiar everyday example of an alternative is the 'time taken' metric, where we express distances by saying how long it takes us to get there. Like: "The shops are five minutes away from my house." Where we are constrained to move slowly, 'time taken' distances are greater.

Travel along a good road is fast/short; travel across rough countryside is slow/long. We can assign a 'speed' to each point, and then define a new sort of geometry by defining 'straight lines' as the fastest route between points, distances as 'time elapsed' travelling along the route, circles as the set of points that can be reached from the centre point in a fixed time, and so on. If we set the speed to be a constant everywhere, we get flat Euclidean geometry. If the speed varies from place to place, it is not.

So one way of thinking about non-Euclidean geometry is to embed the surface in a higher-dimensional space, and use the Euclidean metric in that embedding space to specify the non-Euclidean metric in the embedded space. We embed a curved 2D sphere in a flat 3D geometry, and distances between points on the sphere are inherited from the Euclidean geometry they are embedded in. Most of our language about curvature and many of our intuitions arise from this embedding paradigm.

But another way is to instead specify a different 'distance' between points in our pure 2D space, and this is the way the mathematicians have gone. In particular, this allows us to use functions that give zero or negative squared distances between points, such as the Minkowski geometry of special relativity, which are impossible to express by embedding in any sort of Euclidean space.

(We might try to imagine this in our 'time taken' metric if there were places where movement in certain regions was instantaneous, allowing a zero 'distance' between distinct points, or even negative, travelling backwards in time. Physically unrealistic, of course, but it is at least imaginable.)

If surfaces are topologically the same, we can map any metric from one to the other. So for example, if we take a sphere with one point removed, map pairs of points stereographically onto a plane, and define the distance between the points on the sphere to be the Euclidean distance between the mapped points on the plane, we end up with a sphere with a flat Euclidean metric (except at one point). And vice versa, of course.

Curvature then is a way to measure how non-Euclidean our alternative metric is. If it is harder to travel through the middle than around the outside, compared to Euclidean space, then it has positive curvature. If it is harder to travel around the outside, then it has negative curvature. Or for a more precise definition, if we take a circle (defined as points at a fixed 'distance' from a central point) and compare its radius to its circumference, we find the circumference is shorter than $2\pi r$ with positive curvature, and longer than $2\pi r$ with negative curvature. If we divide the difference by $\pi r^3$ and take the limit as $r \rightarrow 0$ we get the Gaussian curvature of the surface.

The early theory in non-Euclidean geometry routinely used the embedding paradigm, but the mathematicians eventually became dissatisfied with it because it mixed up the intrinsic geometry of the surface itself (based entirely on distances and angles within the surface) with the extrinsic geometry of the arbitrary embedding, and they could not be sure whether they were being misled by intuitions arising from how they had chosen to embed the surface. The metric formalism stops them doing that. So while the embedding paradigm is still useful for beginners, to build intuition, you are encouraged to stop using it as soon as you can.

There is no necessity for an embedding space with extra dimensions to enable curvature, and there are plenty of non-Euclidean geometries that cannot be so embedded. The alternative is to redefine distances between points to follow a different rule to the Euclidean definition.

The topology and metric of a surface are defined separately. We can define a particular topology first - flat plane, sphere, torus, Mobius strip,... - and then define a metric on it. The metric determines the curvature. A sphere cannot be mapped smoothly onto a plane because they have different topologies, but this is not because of the curvature.