<p>It is quite possible for a manifold to be bounded and finite and not have a geometric centre (that is on the manifold)</p>
<p>To talk about an &quot;arithmetic mean&quot; you need a way to add up the points.  That requires you to embed the manifold. Now, n-Manifolds may be embedded in <span class="math-container">${\bf R}^{2n}$</span>, and so the best you can say is that the &quot;arithmetic mean&quot; is a point in <span class="math-container">${\bf R}^{2n}$</span>.  A simple example is a 1-manifold circle. It will embed in a plane and the centre of the circle is a point in the plane, and never a point on the circle.</p>
<p>But the embedding is a mathematical construction; it isn't &quot;real&quot;. The manifold exists without any embedding. And so it is correct to say that a 1 dimensional circle doesn't have a centre.</p>
<p>And similarly, a curved, closed three dimension manifold doesn't have a centre (on the manifold).</p>
<p>What this means for the universe, suppose the universe does have positive curvature (That is <strong>not</strong> known) and suppose that the topology of space is a three-sphere, then it does not have a centre.</p>
<p>If we some how embed that three sphere in four dimensional euclidian space, then the universe would be the on the boundary of a four-dimensional disc, and that would have a centre.  But this is a <em>purely mathematical construction</em> and isn't &quot;real&quot;.  There isn't a &quot;real&quot; four-dimensional euclidean space that the universe is <em>really</em> embedded in (or if there is, we have no access to it). The location of that &quot;centre&quot; is entirely dependent on your choice of coordinates. And you would end up with boring triviallities &quot;The centre is at (0,0,0,0).  Where is that?  Nowhere.&quot;  Because a &quot;place&quot; is understood to be a place in the universe, locations outside the universe aren't places.</p>
<p>As for the overall curvature of the universe.  That is still not known to be positive or negative.  Best estimates say it is zero, to within the margin of error, though the effect of dark energy seems to suggest unlimited expansion.</p>

CC BY-SA 4.0
2023-02-25T10:57:58.433Z
I imagine Flatlanders living on te surface of a basketball. They find that their world is curved, as they cannot construct parallel lines and their triangles have angles adding up to more than 180°. But they can't see a center of their world in three-dimensional space.
Oscar Lanzi
CC BY-SA 4.0
2023-02-25T13:33:56.773Z
*" A simple example is a 1-manifold circle. It will embed in a plane and the centre of the circle is a point in the plane, and never a point on the circle."* Sure, but we might still have a centroid on the 1-manifold itself. Eg, on a clock, if we have points of equal mass at 1, 2 & 3 o'clock, the centroid is 2 (but also its antipode at 10). OTOH, if we have n equally spaced points of equal mass, then there's no centroid (or every point is a centroid).
PM 2Ring
CC BY-SA 4.0
2023-02-25T13:42:26.510Z
How do we know that the centroid in this example is at 2 o'clock? Not 8 o'clock?
Oscar Lanzi
CC BY-SA 4.0
2023-02-25T15:33:49.890Z
@Oscar Oops! In my example I should have said 8 o'clock rather than 10 o'clock. But we can (often) eliminate the antipode if we define the distance on the clock to be the shorter (absolute) distance.
PM 2Ring
CC BY-SA 4.0
2023-02-26T00:36:21.923Z
Would it then be true to say that, given a surface with positive curvature, there will be a centroid, extrinsic to that surface? (i.e., that there should exist in 3-dimensional Euclidean space a point that represents the locus of the intersection of all lines perpendicular to the surface, and that this point represents the centroid of the volume enclosed by the surface. My goal is to find an expression for the center of the Universe, if it exists, given that recent findings claim that the Universe has a small but significant positive curvature.)
John Freeman
CC BY-SA 4.0
2023-02-26T07:21:43.293Z
I've edited.  You can choose coordinates, embed and get an expression for the centre. It is at (0,0,0,0) (in my choice of coordintes.  So what, that isn't a place.
James K
CC BY-SA 4.0
2023-02-26T19:17:28.690Z
I can imagine a 2d world which has a center if you count singularities - imagine surface of two water drops connected with their points. The problem Is that the singularity prevent you from drawing a straight line from one drop to the other.
Leos Ondra
CC BY-SA 4.0
2023-02-25T06:48:48.030Z
2023-02-26T07:19:57.150Z
2023-02-26T07:19:57.150Z
James K
<p>Based on the recent Planck Legacy 2018 release confirming the presence of an enhanced lensing amplitude in the
cosmic microwave background (CMB) power spectra compared to that predicted by the standard ΛCDM model, several
authors (Di Valentino et al. (2022), Efstathiou &amp; Gratton (2020), Handley (2019)) have proposed that the Universe
is spatially closed, with a positive curvature . A positive curvature of the Universe presents some tension with the
standard model, which Di Valentino et al. (2022) couch in terms of a &quot;possible crisis in cosmology&quot;.</p>
<p>The value of Ω is of interest because it determines the form of the evolution of an expanding universe (Eingorn et al.,
2019). Based on the ΛCDM model, Ω will have a negligible effect on expansion for early
times, but becomes predominate at successively later times. For  &lt; 0 expansion will continue forever. For  = 0
expansion will continue forever, but the rate will approach zero asymptotically as  goes to infinity. Indefinite expansion implies
that the Universe must be unbounded and have an infinite volume, and thus posses no center. For  &gt; 0 expansion will
halt at some time  and reverse itself due to the inexorable tug of gravity. This implies that the Universe would have a
finite volume. It is claimed that a bounded surface with a finite volume has a geometric center (centroid) which is the arithmetic mean
of all points on the surface (Lovett, 2019).</p>
<p>Is this claim true?</p>
<p>References:</p>
<p>Di Valentino E., Melchiorri A., Silk J., 2022, Nature Astronomy, 4, 196</p>
<p>Efstathiou G., Gratton S., 2020, MNRAS, arXiv:2002.06892v1</p>
<p>Handley W., 2019, arXiv e-prints, arXiv:1908.09139</p>
<p>Eingorn M., Yukselci A. E., Zhouk A., 2019, Eur. Phys. J., 79</p>
<p>Lovett S. J., 2019, Differential Geometry of Manifolds. Barnes and Noble, New York, N. Y.</p>

CC BY-SA 4.0
2023-02-25T08:52:39.897Z
2d analogy Is surface of sphere (imagine Earth). Is has finite area, positive curvature (which can be measured by Flatlanders without leaving the surface). But IT had no center located in this 2d world.
Leos Ondra
<astrophysics>
If the Universe has a positive curvature and a closed spatial geometry, does it therefore have a finite volume and a geometric center?
John Freeman
# If the Universe has a positive curvature and a closed spatial geometry, does it therefore have a finite volume and a geometric center?

(Asked by: username_0 on 2023-02-25)

Based on the recent Planck Legacy 2018 release confirming the presence of an enhanced lensing amplitude in the
cosmic microwave background (CMB) power spectra compared to that predicted by the standard ΛCDM model, several
authors (Di Valentino et al. (2022), Efstathiou & Gratton (2020), Handley (2019)) have proposed that the Universe
is spatially closed, with a positive curvature . A positive curvature of the Universe presents some tension with the
standard model, which Di Valentino et al. (2022) couch in terms of a "possible crisis in cosmology".

The value of Ω is of interest because it determines the form of the evolution of an expanding universe (Eingorn et al.,
2019). Based on the ΛCDM model, Ω will have a negligible effect on expansion for early
times, but becomes predominate at successively later times. For < 0 expansion will continue forever. For = 0
expansion will continue forever, but the rate will approach zero asymptotically as goes to infinity. Indefinite expansion implies
that the Universe must be unbounded and have an infinite volume, and thus posses no center. For > 0 expansion will
halt at some time and reverse itself due to the inexorable tug of gravity. This implies that the Universe would have a
finite volume. It is claimed that a bounded surface with a finite volume has a geometric center (centroid) which is the arithmetic mean
of all points on the surface (Lovett, 2019).

Is this claim true?

References:

Di Valentino E., Melchiorri A., Silk J., 2022, Nature Astronomy, 4, 196

Efstathiou G., Gratton S., 2020, MNRAS, arXiv:2002.06892v1

Handley W., 2019, arXiv e-prints, arXiv:1908.09139

Eingorn M., Yukselci A. E., Zhouk A., 2019, Eur. Phys. J., 79

Lovett S. J., 2019, Differential Geometry of Manifolds. Barnes and Noble, New York, N. Y.

**Tags:** astrophysics
> **Comments:**
> **username_1**: 2d analogy Is surface of sphere (imagine Earth). Is has finite area, positive curvature (which can be measured by Flatlanders without leaving the surface). But IT had no center located in this 2d world.  

---

## 1 answers

### Answer by username_2 on 2023-02-25. Score: 12

It is quite possible for a manifold to be bounded and finite and not have a geometric centre (that is on the manifold)

To talk about an "arithmetic mean" you need a way to add up the points. That requires you to embed the manifold. Now, n-Manifolds may be embedded in ${\bf R}^{2n}$, and so the best you can say is that the "arithmetic mean" is a point in ${\bf R}^{2n}$. A simple example is a 1-manifold circle. It will embed in a plane and the centre of the circle is a point in the plane, and never a point on the circle.

But the embedding is a mathematical construction; it isn't "real". The manifold exists without any embedding. And so it is correct to say that a 1 dimensional circle doesn't have a centre.

And similarly, a curved, closed three dimension manifold doesn't have a centre (on the manifold).

What this means for the universe, suppose the universe does have positive curvature (That is **not** known) and suppose that the topology of space is a three-sphere, then it does not have a centre.

If we some how embed that three sphere in four dimensional euclidian space, then the universe would be the on the boundary of a four-dimensional disc, and that would have a centre. But this is a *purely mathematical construction* and isn't "real". There isn't a "real" four-dimensional euclidean space that the universe is *really* embedded in (or if there is, we have no access to it). The location of that "centre" is entirely dependent on your choice of coordinates. And you would end up with boring triviallities "The centre is at (0,0,0,0). Where is that? Nowhere." Because a "place" is understood to be a place in the universe, locations outside the universe aren't places.

As for the overall curvature of the universe. That is still not known to be positive or negative. Best estimates say it is zero, to within the margin of error, though the effect of dark energy seems to suggest unlimited expansion.

> **Comments:**
> **username_3**: I imagine Flatlanders living on te surface of a basketball. They find that their world is curved, as they cannot construct parallel lines and their triangles have angles adding up to more than 180°. But they can't see a center of their world in three-dimensional space.  
> **username_4**: *" A simple example is a 1-manifold circle. It will embed in a plane and the centre of the circle is a point in the plane, and never a point on the circle."* Sure, but we might still have a centroid on the 1-manifold itself. Eg, on a clock, if we have points of equal mass at 1, 2 & 3 o'clock, the centroid is 2 (but also its antipode at 10). OTOH, if we have n equally spaced points of equal mass, then there's no centroid (or every point is a centroid).  
> **username_3**: How do we know that the centroid in this example is at 2 o'clock? Not 8 o'clock?  
> **username_4**: @Oscar Oops! In my example I should have said 8 o'clock rather than 10 o'clock. But we can (often) eliminate the antipode if we define the distance on the clock to be the shorter (absolute) distance.  
> **username_0**: Would it then be true to say that, given a surface with positive curvature, there will be a centroid, extrinsic to that surface? (i.e., that there should exist in 3-dimensional Euclidean space a point that represents the locus of the intersection of all lines perpendicular to the surface, and that this point represents the centroid of the volume enclosed by the surface. My goal is to find an expression for the center of the Universe, if it exists, given that recent findings claim that the Universe has a small but significant positive curvature.)  
> **username_2**: I've edited.  You can choose coordinates, embed and get an expression for the centre. It is at (0,0,0,0) (in my choice of coordintes.  So what, that isn't a place.  
> **username_1**: I can imagine a 2d world which has a center if you count singularities - imagine surface of two water drops connected with their points. The problem Is that the singularity prevent you from drawing a straight line from one drop to the other.  

---

