Manifold (ConceptTopic, 4)

From Hi.gher. Space

Manifolds are shapes formed from a regular shape, where opposite edges are connected either with or without twists.

Representation

The edges that shall be connected are marked by arrows, and the direction of the arrow indicates the orientation of the connection. Edges without arrows are left unconnected.

Square manifolds

These are the best known manifolds.

Cylinder

http://invhost.com/share/cylinder.png

• Note that the cylinder formed this way is actually an uncapped cylinder.

Möbius strip

http://invhost.com/share/m%F6biusstrip.png

• The Möbius strip is the only nonorientable surface that can be embedded in 3D.

Torus

http://invhost.com/share/torus.png

Klein bottle

http://invhost.com/share/kleinbottle.png

• Note that there are two forms of Klein bottle: the Figure-8 shape, and the "ordinary" bottle shape.

This is because there are two ways to "fold up" the shape: you can either make the cylinder first and then get the bottle shape, or you can make the Möbius strip first and then get the Figure-8 shape.

• When immersed in three dimensions, the Klein bottle is self-intersecting.

Real projective plane

http://invhost.com/share/realprojectiveplane.png

• When immersed in three dimensions, the real projective plane is self-intersecting.

This immersion is a combination of the two forms of Klein bottle: you take the figure-8 shape, split it open and insert one end through the side of the other, attaching it on the inside.^(?)

Notes

• There is no manifold for a sphere. This is because a sphere has a point of convergance, and if you go off the top of a sphere, you end up going down it again, which cannot be defined by the manifold representations. Similarly, there is no manifold for any 3D shape with a genus of zero.