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Mathematician here. ItIt is very common for manifolds to arise that do not embed in Euclidean space in any obvious (or particularly natural) way. Indeed, the reason that mathematicians think of manifolds without a surrounding ambient space isn't necessarily Occam's razor -- it's just that it wouldn't be practical to develop the theory of manifolds in a context where an ambient space is always required.

Mathematician here. It is very common for manifolds to arise that do not embed in Euclidean space in any obvious (or particularly natural) way. Indeed, the reason that mathematicians think of manifolds without a surrounding ambient space isn't necessarily Occam's razor -- it's just that it wouldn't be practical to develop the theory of manifolds in a context where an ambient space is always required.

Mathematician here. It is very common for manifolds to arise that do not embed in Euclidean space in any obvious (or particularly natural) way. Indeed, the reason that mathematicians think of manifolds without a surrounding ambient space isn't necessarily Occam's razor -- it's just that it wouldn't be practical to develop the theory of manifolds in a context where an ambient space is always required.

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ThatPart of what's important to understand here is that mathematicians don't use manifolds primarily as a way of describing shapes of physical objects. For a simple example, parabolas are quite important in mathematics, but not because you tend to see that many actual parabolas in everyday life. Instead, parabolas are important for a theoretical reason -- they are graphs of quadratic equations, and studying the graphs helps you understand quadratic equations better. As a general rule, the primary motivation in mathematics for studying geometry isn't to model physical reality directly -- it's because understanding geometry helps us to understand equations and other mathematical objects, which can then in turn be used to model reality. Abstract geometric manifolds like the projective plane and PSL(2,R) are very important in this regard, even if they don't necessarily describe a shape that you're likely to run into in real life.

All that being said, it does turn out that every topological manifold can be embedded into a Euclidean space of sufficiently high dimension --- this is the content of the Whitney embedding theorem. According to this theorem, it's possible to embed the projective plane into four-dimensional Euclidean space, and it's possible to embed PSL(2,R) into six-dimensional Euclidean space. (Actually, PSL(2,R) embeds into three-dimensional Euclidean space -- it turns out to be homeomorphic to an open solid torus.) However, this theorem takes some work to prove, and you shouldn't have to prove such a hard theorem just to get theory of manifolds off the ground.

That being said, it does turn out that every topological manifold can be embedded into a Euclidean space of sufficiently high dimension --- this is the content of the Whitney embedding theorem. According to this theorem, it's possible to embed the projective plane into four-dimensional Euclidean space, and it's possible to embed PSL(2,R) into six-dimensional Euclidean space. (Actually, PSL(2,R) embeds into three-dimensional Euclidean space -- it turns out to be homeomorphic to an open solid torus.) However, this theorem takes some work to prove, and you shouldn't have to prove such a hard theorem just to get theory of manifolds off the ground.

Part of what's important to understand here is that mathematicians don't use manifolds primarily as a way of describing shapes of physical objects. For a simple example, parabolas are quite important in mathematics, but not because you tend to see that many actual parabolas in everyday life. Instead, parabolas are important for a theoretical reason -- they are graphs of quadratic equations, and studying the graphs helps you understand quadratic equations better. As a general rule, the primary motivation in mathematics for studying geometry isn't to model physical reality directly -- it's because understanding geometry helps us to understand equations and other mathematical objects, which can then in turn be used to model reality. Abstract geometric manifolds like the projective plane and PSL(2,R) are very important in this regard, even if they don't necessarily describe a shape that you're likely to run into in real life.

All that being said, it does turn out that every topological manifold can be embedded into a Euclidean space of sufficiently high dimension --- this is the content of the Whitney embedding theorem. According to this theorem, it's possible to embed the projective plane into four-dimensional Euclidean space, and it's possible to embed PSL(2,R) into six-dimensional Euclidean space. (Actually, PSL(2,R) embeds into three-dimensional Euclidean space -- it turns out to be homeomorphic to an open solid torus.) However, this theorem takes some work to prove, and you shouldn't have to prove such a hard theorem just to get theory of manifolds off the ground.

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Mathematician here. It is very common for manifolds to arise that do not embed in Euclidean space in any obvious (or particularly natural) way. Indeed, the reason that mathematicians think of manifolds without a surrounding ambient space isn't necessarily Occam's razor -- it's just that it wouldn't be practical to develop the theory of manifolds in a context where an ambient space is always required.

The simplest examples of manifolds that don't embed naturally into an ambient space are quotients of existing manifolds, i.e. manifolds obtained from other manifolds by identifying (or gluing) together certain sets of points. For example, the projective plane is a surface that can be obtained from a sphere by gluing together every pair of antipodal points. The result is a perfectly good surface (it's just like a sphere, except you only need to go halfway around to get back to where you started), but it's not immediately clear how to embed this surface into Euclidean space. Similarly, the projective special linear group PSL(2,R) is obtained from the special linear group SL(2,R) by identifying pairs of matrices that are negatives of one another. Again, SL(2,R) embeds into R^4 using the entries of the matrices, but there is no immediately obvious embedding of PSL(2,R) into a Euclidean space. Both of these examples are quite important in mathematics --- the projective plane is the simplest non-orientable closed surface and the main setting of plane projective geometry, and PSL(2,R) is a simple Lie group and is the group of all isometries (i.e. symmetries) of the hyperbolic plane.

That being said, it does turn out that every topological manifold can be embedded into a Euclidean space of sufficiently high dimension --- this is the content of the Whitney embedding theorem. According to this theorem, it's possible to embed the projective plane into four-dimensional Euclidean space, and it's possible to embed PSL(2,R) into six-dimensional Euclidean space. (Actually, PSL(2,R) embeds into three-dimensional Euclidean space -- it turns out to be homeomorphic to an open solid torus.) However, this theorem takes some work to prove, and you shouldn't have to prove such a hard theorem just to get theory of manifolds off the ground.

In addition, there are some other contexts than topology where embedding theorems are either very difficult or impossible to obtain. For example, if you consider Riemannian manifolds (the main objects of study in differential geometry), it can be very difficult to show that there's an isometric embedding of a given Riemannian manifold into a Euclidean space (see the Nash embedding theorems). For example, the hyperbolic plane is very easy to define, but it's very hard to find a smooth isometric embedding of it into a Euclidean space, and I think it's still an open question whether there exists an analytic embedding (i.e. an embedding defined by a single power series formula).

Mathematician here. It is very common for manifolds to arise that do not embed in Euclidean space in any obvious (or particularly natural) way. Indeed, the reason that mathematicians think of manifolds without a surrounding ambient space isn't necessarily Occam's razor -- it's just that it wouldn't be practical to develop the theory of manifolds in a context where an ambient space is always required.

The simplest examples of manifolds that don't embed naturally into an ambient space are quotients of existing manifolds, i.e. manifolds obtained from other manifolds by identifying (or gluing) together certain sets of points. For example, the projective plane is a surface that can be obtained from a sphere by gluing together every pair of antipodal points. The result is a perfectly good surface, but it's not immediately clear how to embed this surface into Euclidean space. Similarly, the projective special linear group PSL(2,R) is obtained from the special linear group SL(2,R) by identifying pairs of matrices that are negatives of one another. Again, SL(2,R) embeds into R^4 using the entries of the matrices, but there is no immediately obvious embedding of PSL(2,R) into a Euclidean space. Both of these examples are quite important in mathematics --- the projective plane is the simplest non-orientable closed surface and the main setting of plane projective geometry, and PSL(2,R) is a simple Lie group and is the group of all isometries (i.e. symmetries) of the hyperbolic plane.

That being said, it does turn out that every topological manifold can be embedded into a Euclidean space of sufficiently high dimension --- this is the content of the Whitney embedding theorem. According to this theorem, it's possible to embed the projective plane into four-dimensional Euclidean space, and it's possible to embed PSL(2,R) into six-dimensional Euclidean space. (Actually, PSL(2,R) embeds into three-dimensional Euclidean space -- it turns out to be homeomorphic to an open solid torus.) However, this theorem takes some work to prove, and you shouldn't have to prove such a hard theorem just to get theory of manifolds off the ground.

In addition, there are some other contexts than topology where embedding theorems are either very difficult or impossible to obtain. For example, if you consider Riemannian manifolds (the main objects of study in differential geometry), it can be very difficult to show that there's an isometric embedding of a given Riemannian manifold into a Euclidean space (see the Nash embedding theorems). For example, the hyperbolic plane is very easy to define, but it's very hard to find a smooth isometric embedding of it into a Euclidean space, and I think it's still an open question whether there exists an analytic embedding (i.e. an embedding defined by a single power series formula).

Mathematician here. It is very common for manifolds to arise that do not embed in Euclidean space in any obvious (or particularly natural) way. Indeed, the reason that mathematicians think of manifolds without a surrounding ambient space isn't necessarily Occam's razor -- it's just that it wouldn't be practical to develop the theory of manifolds in a context where an ambient space is always required.

The simplest examples of manifolds that don't embed naturally into an ambient space are quotients of existing manifolds, i.e. manifolds obtained from other manifolds by identifying (or gluing) together certain sets of points. For example, the projective plane is a surface that can be obtained from a sphere by gluing together every pair of antipodal points. The result is a perfectly good surface (it's just like a sphere, except you only need to go halfway around to get back to where you started), but it's not immediately clear how to embed this surface into Euclidean space. Similarly, the projective special linear group PSL(2,R) is obtained from the special linear group SL(2,R) by identifying pairs of matrices that are negatives of one another. Again, SL(2,R) embeds into R^4 using the entries of the matrices, but there is no immediately obvious embedding of PSL(2,R) into a Euclidean space. Both of these examples are quite important in mathematics --- the projective plane is the simplest non-orientable closed surface and the main setting of plane projective geometry, and PSL(2,R) is a simple Lie group and is the group of all isometries (i.e. symmetries) of the hyperbolic plane.

That being said, it does turn out that every topological manifold can be embedded into a Euclidean space of sufficiently high dimension --- this is the content of the Whitney embedding theorem. According to this theorem, it's possible to embed the projective plane into four-dimensional Euclidean space, and it's possible to embed PSL(2,R) into six-dimensional Euclidean space. (Actually, PSL(2,R) embeds into three-dimensional Euclidean space -- it turns out to be homeomorphic to an open solid torus.) However, this theorem takes some work to prove, and you shouldn't have to prove such a hard theorem just to get theory of manifolds off the ground.

In addition, there are some other contexts than topology where embedding theorems are either very difficult or impossible to obtain. For example, if you consider Riemannian manifolds (the main objects of study in differential geometry), it can be very difficult to show that there's an isometric embedding of a given Riemannian manifold into a Euclidean space (see the Nash embedding theorems). For example, the hyperbolic plane is very easy to define, but it's very hard to find a smooth isometric embedding of it into a Euclidean space, and I think it's still an open question whether there exists an analytic embedding (i.e. an embedding defined by a single power series formula).

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