non orientable surface

Non orientable phenomena are a consequence about the consideration of the tangent bundles regarding an embeddingMathworldPlanetmath. One asks if e:AB is an embedding then how the tangent bundles TA and TB relate?

For example: we could consider the core (simple close curve) of an cylinder S1×I or in a Mobius band Mo¨. First we can observe that if C1=S1×{12} has as a regularPlanetmathPlanetmathPlanetmathPlanetmath neighborhoodMathworldPlanetmathPlanetmath whose boundary is two componentPlanetmathPlanetmathPlanetmath disconnected curve (in fact two disjoint circles), while the boundary of a regular neighborhood N of the core curve Co¨ is a single circle: Mo¨.

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 Co¨ which after a full turn we are going to find a reflexion of the normal axe.

Now employing the standard classification of closed surfaces we will construct another kind.

These are the only types of orientable surfaces: O0 the sphere; O1 the two torus; O2=O1#O1 the bitoro; O3=O1#O1#O1 the tritoro,… etc, i.e.


So, with the connected sumMathworldPlanetmathPlanetmath device we have:

The projective planeMathworldPlanetmath

P2 = (O0intD)Mo¨
= DMo¨

The Klein bottle

P2#P2 = [O0(intD1intD2)][(Mo¨)1(Mo¨)2]
= (Mo¨)1(Mo¨)2

If we standarize as N1=P2 and N2=P2#P2, then the genus three non orientable surface is

N3 = P2#P2#P2
= N2#P2
= O1#P2
= ([O0(intD1intD2intD3)][(Mo¨)1(Mo¨)2(Mo¨)3]
= (O1intD)Mo¨
= (N2intD)Mo¨


(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
Canonical name NonOrientableSurface
Date of creation 2013-03-22 19:02:29
Last modified on 2013-03-22 19:02:29
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 33
Author juanman (12619)
Entry type Definition
Classification msc 53A05
Classification msc 57M20
Classification msc 14J29
Related topic Surface