Is There a Closed Surface in 3 Space That is Not Curved?

[hamsterman]

I've been watching http://www.youtube.com/watch?v=Pm5ROyoaMZA" on youtube where it was shown that a cone has curved space (from about 1:05:00). Using the same method of parallel transportation it seems to me that every 3d shape, other than a plane has some points of curvature. However the same lecture states that a cylinder can be easily flattened and has no curvature. Am I dong something wrong, or did they ignore the top/bottom of the cylinder?

 

[Rasalhague]

You're right that what he calls a cylinder (around 01:32) doesn't include the top and bottom, only the tube that can be unwrapped and flattened out to make a square. People use the word cylinder in a few different ways, so if you meet it elsewhere it might have another meaning.

http://mathworld.wolfram.com/Cylinder.html

Imagine transporting a tangent vector (visualised as an arrow) in a loop around the kind of cylinder he's talking about. However you draw the loop, the vector will be pointing in the same direction when you get back to where you started. If a space has this property, he defines it as "flat". If not, the space is "curved" somewhere inside the loop.

You might be misunderstanding what he means by 2d and 3d. When he talks about a plane, a cone, a cylinder, a torus, a sphere, etc. he's thinking of each of these shapes only as a two dimensional surface. Because we think of ourselves instinctively as living in three dimensional space, we find it easiest to visualise these 2d shapes as embedded in three dimensions, but mathematically, the shapes he talks about in this part of the lecture are still considered to be 2d shapes. If I've understood this right, the geometric properties that he's talking about don't depend on how we choose to visualise them.

 

[hamsterman]

ok, thanks

 

[mugaliens]

People use the word cylinder in a few different ways, so if you meet it elsewhere it might have another meaning.

I use it to mean a circle elongated along it's axis. What other meaning is there?

 

[HallsofIvy]

Any closed curve, in the some plane, with all point "elongated" along parallel lines, not in that plane but not necessarily perpendicular to that plane, is a cylinder. If the closed curve is a circle, it is a "circular cylinder". If the lines are perpendicular to the plane, it is a "right cylinder". The thing you call a "cylinder" is properly called a "right circular cylinder".

 

[wofsy]

I've been watching http://www.youtube.com/watch?v=Pm5ROyoaMZA" on youtube where it was shown that a cone has curved space (from about 1:05:00). Using the same method of parallel transportation it seems to me that every 3d shape, other than a plane has some points of curvature. However the same lecture states that a cylinder can be easily flattened and has no curvature. Am I dong something wrong, or did they ignore the top/bottom of the cylinder?

There are many surfaces that can be flattened onto a plane. The cylinder is only one. A Mobius strip made of paper is another as are helicoids and other ruled surfaces. the key insight is that bending is not the same as curving. A cylinder is bent but not curved.

Try to prove that there is no closed surface in 3 space that is not curved somewhere. This is a theorem of Hilbert and is easy.

One would think that a double cylinder, a torus, could also made be bent but not curved but Hilbert's theorem says this is impossible in 3 space. However, in 4 space it easy to make the flat torus. Try writing down an equation for a flat torus embedded in a sphere of radius one centered at the origin of 4 space.