Can Curvature Only Be Formed by Removing Sections of Flat Space?

[friend]

I wonder if curvature necessarily means space has been removed. The typical example is forming a "curved" surface by cutting out a triangle from a flat surface, and then gluing the remaining side back together. This forms of a cone which is a type of curved surface. What is the generalization of this? Does this mean that curvature can only be formed by cutting out sections of a flat space? Thanks.

 

[HallsofIvy]

Where did you get that idea? Any surface has "curvature" (a plane has curvature "0" at every point- any non-plane surface has non-zero curvature). A sphere, for example has curvature 1/R, where R is the radius at any point. An ellipse has non-constant curvature.

 

[micromass]

I wonder if curvature necessarily means space has been removed. The typical example is forming a "curved" surface by cutting out a triangle from a flat surface, and then gluing the remaining side back together. This forms of a cone which is a type of curved surface. What is the generalization of this? Does this mean that curvature can only be formed by cutting out sections of a flat space? Thanks.

A cone actually has zero curvature, at least if you talk about intrinsic curvature. It does have nonzero extrinsic curvature.

 

[WWGD]

A sphere has constant non-zero curvature.

 

[Chronos]

You can't remove a piece of nothing. Space is not a 'fabric', or otherwise substantive.

 

[WWGD]

There are a lot of different types of curvature, maybe you could specify which one you mean.

 

[homeomorphic]

Actually, if you take pieces of a flat space and do something like that--that is, you bend them without stretching, you never get anything but a flat space in the sense that the intrinsic geometry of the space is that of a flat space. For example, a sphere has a geometry in which triangles have angles that add up to more than 180 degrees, in contrast to Euclidean geometry. If you want to transform a flat space that has Euclidean geometry into a curved one that has spherical geometry, you would have to do more than bend. You would have to stretch. You can't do that with paper because paper tears if you try to stretch it.

 

[friend]

By cutting out space or adding it in, I think I meant "stretching" the rest together, or pushing the rest apart to make room. This in effect squeezes or stretches the surrounding space and with it the metric. This stretching or squeezing of space would change the metric from being flat and so create curvature, right?

 

[homeomorphic]

Stretching is necessary, but not sufficient. You could just scale everything up by a constant factor, but that wouldn't create any curvature, for example.

 

[WWGD]

By stretching or doing some transformations, if your object is embedded, a change in the embedding changes t he subspace/"induced " pullback metric. But tearing is not continuous.