2-sphere manifold intrinsic definition

[cianfa72]

TL;DR Summary

About the 2-sphere manifold intrinsic definition without looking at its embedding in $\mathbb R^3$

Hi,
in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space $\mathbb R^3$.

On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts.

I believe the same view might also be applied to the 2-sphere by starting from two charts (i.e. two copies of $\mathbb R^2$) and defining their "gluing" instructions (i.e. their transition maps).

Does the above make sense?

 

[Bosko]

You can do it in two steps
First make a (surface of the) cube
Then stretch/squeeze radially to became a sphere

 

[cianfa72]

You can do it in two steps
First make a (surface of the) cube
Then stretch/squeeze radially to became a sphere

Yes, the problem is that it is an "extrinsic" view that employ the embedding in $\mathbb R^3$.

 

[martinbn]

Why do you want that?

The two sphere $S^2$ is the same as the projective line over the complex numbers $\mathbb CP^1$, which is defined the way you want as a quotient.

 

[cianfa72]

The two sphere $S^2$ is the same as the projective line over the complex numbers $\mathbb CP^1$, which is defined the way you want as a quotient.

Ah ok, you mean define $S^2$ as the quotient space $\mathbb CP^1$ on $\mathbb C^2$ w.r.t. the equivalence relation $\{ y \sim x \text{ } | y = kx, x,y \neq 0, x,y \in \mathbb C^2 \}$.

Edit: consider the pair $x=(x_1,x_2), x_1,x_2 \in \mathbb C^2$. When $x_1 \neq 0$ we get $[x]=[x_2/x_1, 1]$ hence $x_2/x_1$ defines a chart for the manifold. Now the equivalence class for points $x$ with $x_1=0$ should be mapped to the point at infinity in that chart.

 

[fresh_42]

How about $\mathbb{S}^2 = \operatorname{SO}(3,\mathbb{R}) \big / \operatorname{SO}(2,\mathbb{R}).$ You can decide whether you consider the orthogonal group as rotations, an algebraic variety, a topological group, a manifold in $\mathbb{R}^{n^2-1}$, or whatever. It even allows you to regard $\mathbb{S}^2$ as last term in a short-exact sequence of groups.

 

[cianfa72]

The point I was making is that, if one has an atlas for a manifold, then one can uniquely "glue" the pieces of open sets in $\mathbb R^n$ according the chart's transition maps in order to "build" the manifold.

 

[fresh_42]

The point I was making is that, if one has an atlas for a manifold, then one can uniquely "glue" the pieces of open sets in $\mathbb R^n$ according the chart's transition maps in order to "build" the manifold.

So?

Then consider the Riemann projection of the 2-sphere, forget about the embedding, and end up with $\mathbb{C}P^1$ as mentioned above with 2 charts.
Summary: https://de.wikipedia.org/wiki/Riemannsche_Zahlenkugel
Details: https://en.wikipedia.org/wiki/Riemann_sphere

 

[mathwonk]

note that any construction that gives a compact simply connected 2-manifold will do.

 

[mathwonk]

you asked for an intrinsic approach. two intrinsic properties that the 2 -sphere possesses are "compactness" and "simply connectedness". I.e. every covering of the 2-sphere by a family of open sets can be reduced to a cover by a finite number of those sets, and every closed loop on the sphere can be shrunk to a point on the surface of the sphere. The 2-sphere is the only 2-manifold with both those properties. Thus if someone gives you a construction claiming to give the 2-sphere, you can prove it is indeed the 2-sphere by showing it is a 2-manifold, and it is both compact and simply connected.
This may or may not be easier than finding an actual homeomorphism with the usual 2-sphere. Subsets of Euclidean space are compact precisely when they are both closed and bounded. Continuous images of compact spaces are also compact. Finite unions of compact spaces are compact. Convex subsets of euclidean space are simply connected.

To me the simplest chart construction of the 2-sphere is to appropriately identify the boundary circles of two closed discs, or thicken the boundaries a bit to get open charts. This is the fresh_42 approach. Compactness is fairly easy as above, and for simple connectedness, you might need the ability to approximate any continuous loop by one that is made of a finite number of "geodesics", hence essentially lies in one "disc", which is (homeomorphic to) a convex set. Maybe here an actual homeomorphism to the usual 2-sphere is actually easier, and then no need to check simple connectedness.

 

[cianfa72]

To me the simplest chart construction of the 2-sphere is to appropriately identify the boundary circles of two closed discs, or thicken the boundaries a bit to get open charts.

As far as I can understand, you mean take two open discs and "glue" together their ticken boundaries (such glued open regions are actually open annulus from the boundary of each disc).

In this sense the two discs are two open charts.

 

[mathwonk]

actually van Kampen's theorem implies that any space which is the union of two open discs which have open arc-connected intersection, is simply connected. so that does it immediately.

 

[lavinia]

The point I was making is that, if one has an atlas for a manifold, then one can uniquely "glue" the pieces of open sets in $\mathbb R^n$ according the chart's transition maps in order to "build" the manifold.

I see your point as saying that the intrinsic construction of the manifold is obtained solely from its topology i.e. its collection of open sets by gluing them together along overlaps. All other definitions use external topological spaces.

On the other hand external spaces might not be spaces in which the manifold is embedded. So it seems that the idea of extrinsic versus intrinsic from your point of view is ambiguous.

The idea of intrinsic as you have defined it has been important in mathematics if one generalizes it to include properties of the space can be determined solely from its open sets and do not require an external topological space. For instance as @mathwonk pointed out, the Euler characteristic of a manifold is intrinsic in this sense. Sometimes one can even define a manifold from these intrinsic properties. For instance, the topology of a closed orientable surface is determined by its Euler characteristic. (This though is not true in higher dimensions. )

If I were to take guess, the idea of external originated in the early days of differential geometry when surfaces were viewed as subsets of 3 space and the abstract notion of manifold hadn't been discovered yet.

 

[cianfa72]

All other definitions use external topological spaces.

On the other hand external spaces might not be spaces in which the manifold is embedded.

Take for instance the figure 8 that is immersed in $\mathbb R^2$, however there is no embedding in it.

So it seems that the idea of extrinsic versus intrinsic from your point of view is ambiguous.

As far as I can understand, you mean the extrinsic view is ambiguous since there are different "ambient" spaces where the "intrinsic" manifold can be immersed or embedded.

 

[lavinia]

As far as I can understand, you mean the extrinsic view is ambiguous since there are different "ambient" spaces where the "intrinsic" manifold can be immersed or embedded.

No. I mean any situation where external topologies are used to define the manifold. For instance defining the sphere as a quotient space of $SO(3)$. Except for embeddings these are all taken to be intrinsic while any definition that uses an embedding is by definition extrinsic even though it also uses external topologies.

I don't understand your point about the figure 8.

 

[fresh_42]

If I were to take guess, the idea of external originated in the early days of differential geometry when surfaces were viewed as subsets of 3 space and the abstract notion of manifold hadn't been discovered yet.

I would date differential geometry's birth to Riemann's habilitation speech in 1854 "About the Hypotheses that Underly the Geometry". Gauß had selected it among three suggestions from Riemann. Gauß had already introduced curvature. However, Riemann's speech was held in front of a non-mathematical audience, so chances are high that it was mainly about embedded surfaces. It hasn't been published before 1866. The early calculations were all in local, curved coordinates (Gauß, 1827) so it is cumbersome to distinguish whether this was embedded or on charts. The idea of charts was in the world by 1827. Dieudonné prefers the distinction between differentiable manifolds and Riemannian geometry, depending on whether $ds^2$ is involved or not. Tensor calculus came later (Ricci, 1887).

 

[cianfa72]

I don't understand your point about the figure 8.

The figure 8 (lemniscate) is 1D "intrinsic" manifold that can be immersed in $\mathbb R^2$, however the map that gives the immersion of the open interval $(-\pi, \pi)$ in $\mathbb R^2$ is not an homeomorphism on its image (simply because the $\mathbb R^2$ subspace topology is different from the domain open set topology).

 

[cianfa72]

No. I mean any situation where external topologies are used to define the manifold. For instance defining the sphere as a quotient space of $SO(3)$. Except for embeddings these are all taken to be intrinsic while any definition that uses an embedding is by definition extrinsic even though it also uses external topologies.

Ah ok, by external topologies you mean the topology of the space within the manifold is actually "extrinsically" defined (as subset endowed with subspace topology from it).

 

[lavinia]

The figure 8 (lemniscate) is 1D "intrinsic" manifold that can be immersed in $\mathbb R^2$, however the map that that realize the immersion of the open interval $(-\pi, \pi)$ in $\mathbb R^2$ is not an homeomorphism on its image (simply because the $\mathbb R^2$ subspace topology is different from the open set topology).

ok I get it now. An example that also seems ambiguous is a vector bundle over a manifold. The manifold can be defined as the quotient space of the bundle projection map but it is also naturally embedded in the vector bundle as the set of zero vectors in each fiber.

 

[lavinia]

Ah ok, by external topologies you mean the topology of the space within the manifold is actually "extrinsically" defined (as subset endowed with subspace topology from it).

No. The sphere can be defined as a quotient space of $SO(3)$ . However the topology of $SO(3)$ is separate or external to the topology of the sphere. By the definitions used in this thread, defining it as this quotient space would be intrinsic since it does not involve an embedding.

 

[cianfa72]

No. The sphere can be defined as a quotient space of $SO(3)$ . However the topology of $SO(3)$ is separate or external to the topology of the sphere. By the definitions used in this thread, defining it as this quotient space would be intrinsic since it does not involve an embedding.

Ok, in your example $SO(3)$ topology is external w.r.t. the topology of the 2-sphere. The latter is defined as $SO(3)$ quotient space and since its definition doesn't involve an embedding it counts as intrinsic.

 

[mathwonk]

Since Riemann has come up, I would recall a 3rd way of defining the structure of a manifold; i.e. in addition to being a union of local patches, or a quotient of another manifold, it can be defined as a branched cover of a given manifold, by describing the nature of the branching. This of course was Riemann's approach in his 1851 thesis, thus preceding his 1854 lecture, and was, very briefly, referred to in that later lecture. His descriptions I have read mention branched coverings of regions of the plane, hence cannot give compact examples, but ultimately they are extended to coverings of the sphere. Of course that means the sphere itself needs another description, no doubt the usual one as mentioned by Fresh_42. In that regard, note that a covering by local charts is also a quotient of their disjoint union, so in some sense the first two methods are instances of the "same" technique. Rather a group quotient is a very special example of a quotient map which (I believe) is locally like a product or bundle projection.
I am now motivated to read Riemann's habilitationschrift, thanks Fresh!
.............
wow, after only two pages, it already seems as if Cantor maybe took his start for the concept of set theory from Riemann. I.e., Riemann distinguishes continuous manifolds, which he will discuss here, from discrete manifolds, made up of objects all of some given kind.

Moreover, Riemann himself is harking back to Euclid for a starting point, discussing ways of comparably measuring quantities by moving them onto one another, or more simply, being able to compare them only as smaller or larger, if they lie within one another.

 

[cianfa72]

By the definitions used in this thread, defining it as this quotient space would be intrinsic since it does not involve an embedding.

Just to be clear: defining a manifold via an embedding basically means that the manifold being defined actually "arises" as a subset of the "ambient space" it is embedded in through the subspace topology from it.

 

[mathwonk]

after reading another page, Riemann is trying mainly to explain the concept of higher dimensions, by induction, basically taking products. I.e. a curve is swept out by a continuously moving point, a surface by a continuously moving curve, etc.. Thus if the extent of the movement is given by a number, we have a single coordinate on each of the moving curves e.g. sweeping out a surface.
Conversely, given a surface, he envisions a single valued function on the surface, such that setting it equal to any constant value, defines one of the moving curves sweeping out the surface. Thus he is taking the point in view on a manifold used in Morse theory. This would describe a 2-sphere as the union of say the circles at different constant height (constant latitude?). He mentions that the curves can have a special configuration, presumably such as near a pole, but does not go into it, since the lecture is expository.
Then on each of the moving curves on a surface he envisions a single valued function specifying which moving point one is at on that curve. Thus he is representing his n dimensional manifold numerically by defining n functions on it, which would embed it, perhaps locally if he chooses only n functions, i.e. would define a local chart. His words are not very precise, but very suggestive.

The next section begins discussing how to measure curve segments on a manifold, e.g. in such a way that every segment can measure every other.

 

[mathwonk]

Riemann makes clear that an n manifold looks locally like Euclidean n space, in that points (elements) can be fully described by n continuous functions. As Fresh said, it is not clear whether he is thinking of charts or embeddings, but he is using only n coordinates near every point. If we imagine the 2-sphere embedded in 3-space as usual as x^2+y^2+z^2 = 1, then we can cover the sphere by 6 pairs of open hemispheres n(x>0, x<0, y>0, y<0, z>0, z<0), on each one of which two of the coordinates, i.e. either (y,z) or (x,z) or (x,y), will provide such local coordinates.

Thus we can think of the sphere as embedded by the three functions x,y,z, of which we choose an appropriate two near each point, or we can think of the sphere as covered by those 6 charts, on each of which we only consider two functions, which give a homeomorphism of that chart, and which are smoothly related on overlaps.

Conversely, given a (finite) covering by charts, we can presumably extend each chart homeomorphism to n globally defined smooth functions, and then use all of them to embed the manifold in a big product space. Hence there is little difference between the two approaches, at least for compact manifolds. Riemann moreover speaks of infinite dimensional manifolds, even uncountably infinite dimensional ones, well before Cantor made this precise.

The more Riemann I read, the more I see that he anticipated so much later mathematics; it is almost as if most later mathematicians, even great ones, just took Riemann and worked out the details.

The moral for those of us wanting to do research, or just to understand mathematics, is to try to read the great ones, as well as we can, and think about questions that arise.

As something I just learned by reading a few pages of Riemann's lecture, we recall that Euclid "proved" the SAS congruence theorem by appealing to transformations of the plane carrying one triangle to another with the same measurements. Riemann remarks that of course this sort of isometric transformation of figures is possible also on any surface of constant curvature, a fact I had never realized clearly, although it seems obvious on the sphere.

He makes another more cryptic remark that such transformations are possible even without "bending", on a surface of positive curvature, but not on one of negative curvature. This seems plausible for embedded surfaces, but ....????? Another example of a brief remark by Riemann that may repay a lot of thought.

 

[cianfa72]

If we imagine the 2-sphere embedded in 3-space as usual as x^2+y^2+z^2 = 1, then we can cover the sphere by 6 pairs of open hemispheres n(x>0, x<0, y>0, y<0, z>0, z<0), on each one of which two of the coordinates, i.e. either (y,z) or (x,z) or (x,y), will provide such local coordinates.

Thus we can think of the sphere as embedded by the three functions x,y,z, of which we choose an appropriate two near each point,

Such functions that "realize" the embedding near any point (i.e. in an open neighborhood around it) are actually $x=f(y,z), y=f(x,z), z=f(x,y)$ where $f$ is in the form $$f(a,b) = \sqrt {1 - a^2 - b^2 }$$ defined in the open region $a^2 + b^2 < 1$, right ?

or we can think of the sphere as covered by those 6 charts, on each of which we only consider two functions, which give a homeomorphism of that chart, and which are smoothly related on overlaps.

Such 6 charts are the 6 open hemispheres in the former part, I believe (sometimes I get confused from the chart's domain -- i.e. the open patch on the manifold where the chart's map is defined on vs. chart's target open set in $\mathbb R^n$).

 

[pasmith]

TL;DR Summary: About the 2-sphere manifold intrinsic definition without looking at its embedding in $\mathbb R^3$

Hi,
in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space $\mathbb R^3$.

On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts.

You can define $S^2$ as a quotient $[0,1]^2/\sim$ where $$(x_1,y_1) \sim(x_2,y_2) \Leftrightarrow \begin{cases} x_1 = 0, x_2 = 1,\mbox{ and } y_1 = y_2, \\ x_1 = 1, x_2 = 0,\mbox{ and } y_1 = y_2, \\ y_1 = y_2 = 0, \\ y_1 = y_2 = 1, \quad \mbox{or} \\ x_1 = x_2 \mbox{ and } y_1 = y_2. \end{cases}$$ This equivalence first glues the horizontal edges together to form a cylinder, then glues all of the points at the top and bottom of the cylinder together to form the north and south poles respectively.

 

[cianfa72]

This equivalence first glues the horizontal edges together to form a cylinder, then glues all of the points at the top and bottom of the cylinder together to form the north and south poles respectively.

Sorry, I believe your equivalence relation $\sim$ first glues the vertical edges together to form a vertically laid down cylinder.

Then, starting from this intrinsic definition of the 2-sphere as quotient, one can define an embedding in $\mathbb R^3$.

Btw, in your definition of equivalence relation $\sim$, why there is the last term $x_1=x_2 \text { and } y_1=y_2$ ?

 

[lavinia]

Riemann makes clear that an n manifold looks locally like Euclidean n space, in that points (elements) can be fully described by n continuous functions. As Fresh said, it is not clear whether he is thinking of charts or embeddings, but he is using only n coordinates near every point. If we imagine the 2-sphere embedded in 3-space as usual as x^2+y^2+z^2 = 1, then we can cover the sphere by 6 pairs of open hemispheres n(x>0, x<0, y>0, y<0, z>0, z<0), on each one of which two of the coordinates, i.e. either (y,z) or (x,z) or (x,y), will provide such local coordinates.

Thus we can think of the sphere as embedded by the three functions x,y,z, of which we choose an appropriate two near each point, or we can think of the sphere as covered by those 6 charts, on each of which we only consider two functions, which give a homeomorphism of that chart, and which are smoothly related on overlaps.

Conversely, given a (finite) covering by charts, we can presumably extend each chart homeomorphism to n globally defined smooth functions, and then use all of them to embed the manifold in a big product space. Hence there is little difference between the two approaches, at least for compact manifolds. Riemann moreover speaks of infinite dimensional manifolds, even uncountably infinite dimensional ones, well before Cantor made this precise.

The more Riemann I read, the more I see that he anticipated so much later mathematics; it is almost as if most later mathematicians, even great ones, just took Riemann and worked out the details.

The moral for those of us wanting to do research, or just to understand mathematics, is to try to read the great ones, as well as we can, and think about questions that arise.

As something I just learned by reading a few pages of Riemann's lecture, we recall that Euclid "proved" the SAS congruence theorem by appealing to transformations of the plane carrying one triangle to another with the same measurements. Riemann remarks that of course this sort of isometric transformation of figures is possible also on any surface of constant curvature, a fact I had never realized clearly, although it seems obvious on the sphere.

He makes another more cryptic remark that such transformations are possible even without "bending", on a surface of positive curvature, but not on one of negative curvature. This seems plausible for embedded surfaces, but ....????? Another example of a brief remark by Riemann that may repay a lot of thought.ot on

In my naive opinion, the idea of intrinsic quantities originated with the studies of Euler and Gauss. Euler came up with the idea of Euler characteristic which may have been the first discovered topological invariant. Gauss in proving the Gauss-Bonnet theorem showed that the integral of the Gauss curvature over a closed surface is its Euler characteristic. It was a key demonstration of a connection between global properties of the differential geometry of a manifold and its topology. This theorem was generalized in later mathematics to the theory of characteristic classes, an area of profound research in the 20'th century.

Further Gauss's Theorema Egregium was the first demonstration that the Gauss curvature could be computed only from the inner product of tangent vectors to a surface. This suggested that geometry could be defined intrinsically.

I tend not to agree that Differential Geometry started with Riemann. I guess in a sense, one could say this since the modern formalisms came out of his Habilitation Thesis but geometry was highly developed before Riemann and there was an entire school led by Gaspard Monge in the 18'th century. It was Gauss's research that led to the notions of intrinsic geometry and its relation to underlying topology.

 

[lavinia]

An interesting feedback loop is the question of whether every n-manifold, no matter how initially defined ,could be embedded in Euclidean space of some high enough dimension. If so, then the abstract manifolds though arising in many different ways would not contain a manifold that can not be realized extrinsically in Euclidean space. In some sense there would be nothing new despite the intrinsic definitions. For smooth n -manifolds this turns out to be true and is proved as the Whitney Embedding Theorem.

One might also ask whether Riemannian manifolds(positive definite inner products) can always inherit their differential geometry no matter how defined from an embedding in some Euclidean space. That turns out also to be true and is the Nash Embedding Theorem. So strangley, all Riemannian manifolds can be viewed extrinsically.

For me, a surprising consequence is that every Levi-Civita connection on a Riemannian manifold with positive definite inner products is the same as the standard covariant derivative that an embedded manifold in Euclidean space inherits from the Euclidean directional derivative. I have found this helpful to remember when looking at more abstract definitions of covariant derivatives.

 

[fresh_42]

I tend not to agree that Differential Geometry started with Riemann. I guess in a sense, one could say this since the modern formalisms came out of his Habilitation Thesis but geometry was highly developed before Riemann and there was an entire school led by Gaspard Monge in the 18'th century. It was Gauss's research that led to the notions of intrinsic geometry and its relation to underlying topology.

I actually agree although I said I date the birth of differential geometry with Riemann. This was more a rhetorical remark to pinpoint a date. Achievements in science are - as far as I could see always, but definitely mostly - the result of many previous results. Even Einstein already knew that the speed of light was finite, and Noether explicitly mentioned Lie in her papers of 1918. Riemann personally knew and studied Gauß who already introduced the idea of local coordinates almost three decades earlier. I wouldn't go as far as the Euler characteristic which in my opinion is algebraic topology. Gauß-Bonnet was published in 1848, i.e. only six years before Riemann's habilitation. Gauß-Bonnet is very interesting concerning complex analysis where triangulations play a major role and in my sloppy language: another of the many - whilst brilliant - versions of the fundamental theorem of calculus. Is it geometry or is it analysis? Or is it only a corollary to Gauß's considerations about curved surfaces? All this lies in the eyes of the beholder. What it does show once more, is that science is a road, more than it is the sequence of its milestones.

 

[lavinia]

My guess is that in the 18'th century mathematicians were beginning to question what is meant by a geometry. They certainly were wondering why the Parallel Postulate had never been proved and perhaps questioned the intuition of straightness. In Struik's book on Classical Differential Geometry, Gauss is said to have spoken of "lines of least constraint " on a surface as generalizations of the idea of straightness. If the Euclidean straight line was not sufficient, especially after it was known that there was a plane geometry with infinitely many parallels through a point to a give line, Gauss seems to have been trying to come up with an new idea of straightness. A line of least constant had the property that when viewed as the path of a particle moving at constant speed, its acceleration vector always pointed perpendicular to the surface. In other words, an observer moving along the line would experience no acceleration tangent to the surface and if one imagined that the surface was the only thing he could observe and the embedding was unknowable to him i.e. he was a flatlander on a curved surface so to speak, then he would experience no accelerations and his path would seem inertial. And I would guess that he also imagined that an elastic band constrained only to lie on the surface would release tension until it reached equilibrium and become a line of least constraint. This idea is the idea of a geodesic and seems to be the outcome of inquiry into the nature of straightness in the 18'th century.

 

[fresh_42]

Gauss seems to have been trying to come up with an new idea of straightness.

DYK that Gauß has worked as a cartographer! Non-Euclidean geometry was literally his daily bread!
Our former 10 DM bill:

 

[lavinia]

DYK that Gauß has worked as a cartographer! Non-Euclidean geometry was literally his daily bread!
Our former 10 DM note:

View attachment 347975

View attachment 347976

I didn't know that. Although I once read that when he discovered the non-Euclidean plane geometry,he tried to determine whether it or Euclidean geometry held for space by measuring the sum of the angles of huge triangles on the Earth. I think he decided that he could not measure a triangle large enough to detect a deviation from 180 degrees.

I wonder if he thought that if non-Euclidean geometry were the actual geometry of the universe whether that meant that Newtonian physics could not be correct and was valid only approximately in small regions. That must have been a mind bending possibility to contemplate.

 

[fresh_42]

I think he decided that he could not measure a triangle large enough to detect a deviation from 180 degrees.

Given the size of about 150 kilometers in that little diagram on the bill that triangulates parts of the Kingdom of Hanover (of twice that size) which he has been paid for, he must have known.