# Curvature from holes in space?

1. Jul 20, 2014

### 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.

2. Jul 20, 2014

### 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.

3. Jul 20, 2014

### micromass

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

4. Jul 20, 2014

### WWGD

A sphere has constant non-zero curvature.

5. Jul 20, 2014

### Chronos

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

6. Jul 20, 2014

### WWGD

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

7. Jul 21, 2014

### 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.

8. Jul 21, 2014

### 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?

9. Jul 21, 2014

### 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.

10. Jul 22, 2014

### 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.