non orientable surface
For example: we could consider the core (simple close curve) of an cylinder or in a Mobius band . First we can observe that if has as a regular neighborhood whose boundary is two component disconnected curve (in fact two disjoint circles), while the boundary of a regular neighborhood of the core curve is a single circle: .
In terms of tangent bundles we see that we can choose along the cylinder core a consistent normal in the sense that if this curve is traveled then at the end we have the same basis. In contrast with happens in which after a full turn we are going to find a reflexion of the normal axe.
These are the only types of orientable surfaces: the sphere; the two torus; the bitoro; the tritoro,… etc, i.e.
So, with the connected sum device we have:
The projective plane
The Klein bottle
If we standarize as and , then the genus three non orientable surface is
(0,10)*+R^2=”f”; (13,10)*+TM ¨ o =”e”; (15,0)*+M ¨ o =”m”; \ar@. ”f”;”e”?*!/_2mm/⊂; \ar”e”;”m”?*!/_3mm/p;
|Title||non orientable surface|
|Date of creation||2013-03-22 19:02:29|
|Last modified on||2013-03-22 19:02:29|
|Last modified by||juanman (12619)|