Difference between Manifolds and Non-Euclidean spaces

[mikeeey]

I know that manifolds are topological spaces that locally look like euclidean spaces near each point of and open neighbourhood
And non-euclidean spaces are the curved spaces or simply don't match the 5th euclid's axiomThanks .

 

[ShayanJ]

Usually when people say manifold, they only mean a smooth topological space. They may or may not consider a metric to be present.
But when people say non-Euclidean space, they mean a smooth topological space with a metric which is not Euclidean. So they require the manifold to have a metric and they also want it to be different than the Euclidean metric.

Also I should say that technically, an n-dimensional manifold should be like $\mathbb R^n$ locally, not the corresponding Euclidean space. Because Euclidean space is $\mathbb R^n$ plus a Euclidean metric.

 

[robphy]

Note that Minkowski spacetime:
- is non-Euclidean
... but...
- is not curved
- does satisfy the 5th postulate

 

[mikeeey]

Mathematically an euclidean space is and R^n ( real coordinates space ) with dot product [ in math a space is a set with a structure ( the structure here is the dot product )]
and a manifold is a topolgical space ( R^n with topology )
Then what does non-euclidean space ( mathematically ) consist of ( set and stucture )

 

[micromass]

Mathematically an euclidean space is and R^n ( real coordinates space ) with dot product [ in math a space is a set with a structure ( the structure here is the dot product )]
and a manifold is a topolgical space ( R^n with topology )
Then what does non-euclidean space ( mathematically ) consist of ( set and stucture )

In my experience there is no unique thing which can be called non-Euclidean space. It is mostly a collection of things which appear non-Euclidean in different contexts.
Historically, a non-Euclidean space was something satisfying Euclid's postulates except the parallel postulate. This is only the hyperbolic space (spherical and elliptical geometry does not satisfy all Euclid's postulates except the parallel postulate). So in this case, non-Euclidean space is synomymous with hyperbolic geometry.

But non-Euclidean geometry can also mean spaces with nonzero constant curvature. This would encompass the spheres and hyperbolic spaces. In this case, you are then in the context of Riemannian geometry.

But you can also do Cayley-Klein geometry, where non-Euclidean geometries do not at all arise from smooth manifolds, but rather as certain subspaces of projective space. The non-Euclidean geometries here end up being the usual elliptical geometry, the hyperbolic geometry, but also Minkowski and Galilean geometry.

 

[mathwonk]

I like to think about 2 dimensional geometry, already complicated enough. Although the meaning of “Euclidean geometry” is somewhat agreed upon today, as a result of the work of Hilbert, to me there is some room for interpretation as to what the original axioms of Euclid say. E.g. his axiom on the existence of a line through any two point does not mention uniqueness, and his axiom on the arbitrary extendability of a line segment does not mention whether or not the extended line ever returns to pass over the same points again. If these exceptions are allowed, then great circles on spheres are allowed as lines, as well as closed geodesics (length minimizing arcs) on many other surfaces.

In all cases it seems the primary concept is curvature. Moreover if we want the existence of a transitive group of isometries, as Euclid assumes in his proof of Proposition I.4, (SAS congruence), then the curvature must be everywhere constant. Then to me Euclidean geometry is fundamentally geometry of a surface with constant curvature zero. To be sure, one also obtains surfaces which are only “locally Euclidean”, such as a torus, a cylinder, and since Euclid left somewhat unclear the concept of “separation”, also non orientable surfaces like a Klein bottle and Mobius strip.

Positively curved surfaces occur as the spheres, but these can have different radii, so present an infinite spectrum of surfaces, and in the non orientable case we have the corresponding projective planes. Negatively curved surfaces are provided by hyperbolic planes, again with an infinity of different possible negative radii, as well as surfaces only locally hyperbolic, like compact surfaces of any genus greater than one.

One brings some order from this chaos by the fact that if we require extended lines to be infinite, and never return upon themselves, or more precisely we ask for “simply connectedness”, (and if we assume the completeness axiom of Dedekind, as Euclid and Hilbert do not), there are only three types of simply connected surfaces, the flat Euclidean plane, the positively curved spheres, and the negatively curved hyperbolic planes modeled on the disc. All other connected surfaces of constant curvature are “covered” by these, i.e. are obtained as a quotient space of one of these by a group of isometries acting totally discontinuously. These are all manifolds, but by allowing group actions by discontinuous groups with fixed points one obtains also non manifold spaces, with “cone” points. The book by Conway and Goodman Strauss "On the symmetries of things" apparently treats these.

There is a lovely book by Nikulin and Shafarevich classifying locally Euclidean geometries, “Geometries and groups”.

Apparently in higher dimensions the situation is more complicated since it has been pointed out one can have examples such as Minkowski space, equipped with a non Euclidean metric, but still flat. I was unaware of this.

 

[micromass]

A minkowski type situation already appears in 2D when we equip $\mathbb{R}^2$ with the pseudo-inner product $\langle x,y\rangle = x_1x_2 - y_1y_2$. This makes $\mathbb{R}^2$ in a smooth manifold with pseudo-Riemannian metric. So no, it is not a Riemannian manifold, but in the perspective of Klein's Erlanger program, it does qualify as a non-Euclidean geometry.

 

[lavinia]

I know that manifolds are topological spaces that locally look like euclidean spaces near each point of and open neighbourhood

The use of the word Euclidean space for R^n can be confusing. To say that a manifold looks locally like Euclidean space means that is it locally homeomorphic to R^n. There is no concept of geometry implicit in this definition. In Riemannian geometry, manifolds are topological spaces that are then given a geometry by adding a Riemannian metric. The topological Euclidean space becomes geometrically Euclidean when it is given the usual dot product metric. Then one can talk about angles and lengths and other geometric ideas. One can think of a metric as giving the space an idea of measurement.

And non-euclidean spaces are the curved spaces or simply don't match the 5th euclid's axiom
.

It turns out that any topological manifold that is smooth can be given a Riemannian metric. In fact there are infinitely many possible inequivalent Riemannian metrics and each gives a different geometry to the manifold. For instance an egg and a sphere are both topological spheres but they have different metrics and different geometries.

On a manifold with a geometry, it is almost always not useful to think about whether Euclid's fifth postulate is satisfied or not. This is because there is no easy generalization of the idea of a plane. However, one can test whether a space has Euclidean geometry locally by checking whether the Pythagorean theorem holds for small right triangles or by checking whether the sum on the angles of a triangle is always 180 degrees. Other tests, ones that are often described in Physics, are to see whether the circumference of a small sphere is 2pi times its radius or to see whether the distance between line segments that start out parallel is constant when measured from different points. If a manifold passes the locally Euclidean geometry test at every point, then one gets a generalization of the idea of Euclidean geometry to manifolds that have a different topology than R^n. Examples of locally Euclidean surfaces are the flat torus and the flat Klein bottle.

So a manifold with a Riemannian geometry is not Euclidean if it fails the local Euclidean geometry test. In this case, the manifold has a non-zero Riemann curvature tensor so it is not unfair to say that it is a curved space. One needs to be careful here about what curved means though. For instance a cylinder made out of a piece of paper is not curved in this sense. Also, the Riemann curvature tensor is identically zero for a space that does satisfy the local Euclidean geometry test.

Sometimes the idea of geometry is restricted to manifolds that look the same everywhere. Just like Euclidean space which is geometrically uniform, other spaces can also be geometrically uniform. An inhabitant of such a space would not be able to tell where he is from the local geometry. For surfaces, a requirement for uniformity is constant Gauss curvature. Gauss curvature measures the extent to which a surface fails the local Euclidean geometry test. So an egg would not be such a uniformly curved space since it is more pointy at one end but a sphere would be. For constant negative curvature you can look up the pseudosphere for a nice picture.

It turns out that there is a second plane geometry, the one where there are infinitely many parallels rather than only one(so Euclid's 5'th Postulate does not hold) , and this space considered as a having a Riemannian geometry has constant negative Gauss curvature. So it is accurate to say that this non-Euclidean geometry is a curved space - where by curved you mean non-zero constant Gauss curvature. The usual Euclidean geometry in the plane is not called curved because its Gauss curvature is everywhere zero.

In addition to constant curvature, one may require that the manifold be "homogeneous" which means that there is an isometry (metric preserving homeomorphism) of the manifold into itself that maps any point into any other. For instance the sphere is homogeneous since any point can be rotated into any other. This is a stronger idea than any two locations on the manifold being geometrically indistinguishable. One can ,for instance, even find manifolds of zero curvature that have only finitely many isometries.

 

[zinq]

Usually when people say manifold, they only mean a smooth topological space. They may or may not consider a metric to be present.
But when people say non-Euclidean space, they mean a smooth topological space with a metric which is not Euclidean. So they require the manifold to have a metric and they also want it to be different than the Euclidean metric.

Also I should say that technically, an n-dimensional manifold should be like $\mathbb R^n$ locally, not the corresponding Euclidean space. Because Euclidean space is $\mathbb R^n$ plus a Euclidean metric.

This post is very vague and imprecise in several places.

A manifold is a topological space X for which there exists a fixed nonnegative integer n such that every point has a neighborhood that is homeomorphic to a nonempty neighborhood in the Euclidean space Rⁿ. This is about the topology of X.

When people discuss "non-Euclidean" spaces, they are talking about manifolds that are not geometrically flat in the way that Euclidean spaces are. This is not about the topology of X, but about its geometry.

A topological manifold can be given a smooth structure — thus becoming a smooth manifold — in which case it then makes sense to do calculus on the manifold.

A smooth manifold can be further given a geometrical structure, which is defined by assigning to any two tangent vectors at each point what their dot product is. This enables defining lengths of curves, distances between points, and curvatures of various kinds.

There are many implications for the geometry if you know some specifics about the topology, and also vice versa.

 

[zinq]

This post is very vague and imprecise in several places.

A manifold is a topological space X for which there exists a fixed nonnegative integer n such that every point has a neighborhood that is homeomorphic to a nonempty neighborhood in the Euclidean space Rⁿ. This is about the topology of X.

When people discuss "non-Euclidean" spaces, they are talking about manifolds that are not geometrically flat in the way that Euclidean spaces are. This is not about the topology of X, but about its geometry.

A topological manifold can be given a smooth structure — thus becoming a smooth manifold — in which case it then makes sense to do calculus on the manifold.

A smooth manifold can be further given a geometrical structure, which is defined by assigning to any two tangent vectors at each point what their dot product is. This enables defining lengths of curves, distances between points, and curvatures of various kinds.

There are many implications for the geometry if you know some specifics about the topology, and also vice versa.

Added Wednesday: Also, it is mistaken to say "Euclidean space is ℝⁿ plus a Euclidean metric." The term "Euclidean space" covers all the Rⁿ for any n ≥ 0, and just means the topological product of n copies of the real numbers ℝ. The term "Euclidean space" implies nothing about the metric on Rⁿ; it's just a synonym (without specifying the n).

If you want to refer to the metric, you say explicitly that you are considering Rⁿ with the Euclidean metric.

I just checked Wikipedia, and its article on Euclidean space claims erroneously that Rⁿ implies that it has a "Euclidean structure" — an extraordinarily non-standard phrase that apparently means a Euclidean metric. It is not unusual for someone with a little knowledge to write (part of) a Wikipedia article, thereby confusing thousands of people.