What exactly is a hypersurface?

[Tony Stark]

What is hypersurface

 

[AgentSmith]

Its a surface of something with more than three dimensions. Think of a circle, then a sphere, then a hypersphere. A hypersphere has (at least) one more dimension than a sphere, even though we can't visualize it. The hypersphere's surface is called a hypersurface. You can have a hypercube with a hypersurface. All of these are mathematical objects.

 

[mathman]

In general (mathematical) terms, consider an n-dimensional object. Its surface is called a hypersurface of n-1 dimensions.

 

[Nugatory]

In general (mathematical) terms, consider an n-dimensional object. Its surface is called a hypersurface of n-1 dimensions.

That definition is intuitive but a bit too limiting. For example, the plane $x=17$ is a two-dimensional hypersurface in three-dimensional Euclidean space, but it not the surface of any three-dimensional object.

Mathematically, an n-dimensional hypersurface is an n-dimensional submanifold of an (n+1)-dimensional manifold. Examples include mathman's two-dimensional surface of a three-dimensional sphere; the three-dimensional surfaces of simultaneity (constant t coordinate in a given frame) in four-dimensional space-time; just about any two-dimensional surface, whether curved or flat, in three-dimensional Euclidean space...

 

[mathman]

That definition is intuitive but a bit too limiting. For example, the plane $x=17$ is a two-dimensional hypersurface in three-dimensional Euclidean space, but it not the surface of any three-dimensional object.

Mathematically, an n-dimensional hypersurface is an n-dimensional submanifold of an (n+1)-dimensional manifold. Examples include mathman's two-dimensional surface of a three-dimensional sphere; the three-dimensional surfaces of simultaneity (constant t coordinate in a given frame) in four-dimensional space-time; just about any two-dimensional surface, whether curved or flat, in three-dimensional Euclidean space...

It depends on the definition of "object". If you allow things of infinite extent, like a half space, then the plane is a surface.

 

[Nugatory]

It depends on the definition of "object". If you allow things of infinite extent, like a half space, then the plane is a surface.

Indeed it is.
It's conversations like this one that explain why we need the rigorous definitions as well as the intuitive example-based explanations.