# Thread: What is a surface in Maths and/or Physics?

1. I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

Can this be generalized mathematically as we introduce more (higher) dimensions?

Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

*ie as a limit  2. Originally Posted by geordief I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

Can this be generalized mathematically as we introduce more (higher) dimensions?

Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

*ie as a limit
Here is the rigorous mathematical treatment of the concept called "embedding".  3. I am still not clear if a hypersurface is 2 dimensional or [n-1] dimensional for a n-dimensional space.

In particular in 4-D space-time does one work with 2-D surfaces (for example in calculating curvature by means of parallel transporting of a vector) or is use also made of 3-D surfaces (presumably ,at least in my mind constructed by setting one of the dimensional variables to a constant).

If the 3-D surface does exist and is useful can it be built up successively from the 6(?) combinations of 2-D at any point (event)?  4. Originally Posted by geordief I am still not clear if a hypersurface is 2 dimensional
Only in 3 dimensions

or [n-1] dimensional for a n-dimensional space.
The math is quite clear on the subject. Have you read the web page I linked for you?  5. No but I will now. And I'll get back then......  6. Originally Posted by AndrewC Only in 3 dimensions

The math is quite clear on the subject. Have you read the web page I linked for you?
OK I have read the main body of that link and think I understand that a hypersurface is embedded in the manifold with 1 more dimensions than it has.

Am I right though to think that there are nevertheless also 2D surfaces in higher dimensional manifolds (esp 4D space-time manifolds) even if is incorrect to refer to them as "hypersurfaces".

Are they just referred to as surfaces or 2D surfaces?  7. Originally Posted by geordief OK I have read the main body of that link and think I understand that a hypersurface is embedded in the manifold with 1 more dimensions than it has.

Am I right though to think that there are nevertheless also 2D surfaces in higher dimensional manifolds (esp 4D space-time manifolds) even if is incorrect to refer to them as "hypersurfaces".

Are they just referred to as surfaces or 2D surfaces?
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