non orientable surface

Non orientable phenomena are a consequence about the consideration of the tangent bundles regarding an embedding. One asks if $e:A\to B$ is an embedding then how the tangent bundles $T A$ and $T B$ relate?

For example: we could consider the core (simple close curve) of an cylinder $S^{1}\times I$ or in a Mobius band $M\ddot{o}$ . First we can observe that if $C_{1}=S^{1}\times\{\frac{1}{2}\}$ has as a regular neighborhood whose boundary is two component disconnected curve (in fact two disjoint circles), while the boundary of a regular neighborhood $N$ of the core curve $C\ddot{o}$ is a single circle: $\partial M\ddot{o}$ .

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 $C\ddot{o}$ 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: $O_{0}$ the sphere; $O_{1}$ the two torus; $O_{2}=O_{1}\#O_{1}$ the bitoro; $O_{3}=O_{1}\#O_{1}\#O_{1}$ the tritoro,… etc, i.e.

$O_{g}=O_{1}\#\cdots\#O_{1}$

So, with the connected sum device we have:

The projective plane

	$\displaystyle{\mathbb{R}}P^{2}$	$\displaystyle=$	$\displaystyle(O_{0}\setminus{\rm{int}}D)\cup_{\partial}M\ddot{o}$
		$\displaystyle=$	$\displaystyle D\cup_{\partial}M\ddot{o}$

The Klein bottle

	$\displaystyle{\mathbb{R}}P^{2}\#{\mathbb{R}}P^{2}$	$\displaystyle=$	$\displaystyle[O_{0}\setminus({\rm{int}}D_{1}\sqcup{\rm{int}}D_{2})]\cup_{% \partial}[(M\ddot{o})_{1}\sqcup(M\ddot{o})_{2}]$
		$\displaystyle=$	$\displaystyle(M\ddot{o})_{1}\cup_{\partial}(M\ddot{o})_{2}$

If we standarize as $N_{1}={\mathbb{R}}P^{2}$ and $N_{2}={\mathbb{R}}P^{2}\#{\mathbb{R}}P^{2}$ , then the genus three non orientable surface is

$\displaystyle N_{3}$	$\displaystyle=$	$\displaystyle{\mathbb{R}}P^{2}\#{\mathbb{R}}P^{2}\#{\mathbb{R}}P^{2}$
	$\displaystyle=$	$\displaystyle N_{2}\#{\mathbb{R}}P^{2}$
	$\displaystyle=$	$\displaystyle O_{1}\#{\mathbb{R}}P^{2}$
	$\displaystyle=$	$\displaystyle([O_{0}\setminus({\rm{int}}D_{1}\sqcup{\rm{int}}D_{2}\sqcup{\rm{% int}}D_{3})]\cup_{\partial}[(M\ddot{o})_{1}\sqcup(M\ddot{o})_{2}\sqcup(M\ddot{% o})_{3}]$
	$\displaystyle=$	$\displaystyle(O_{1}\setminus{\rm{int}}D)\cup_{\partial}M\ddot{o}$
	$\displaystyle=$	$\displaystyle(N_{2}\setminus{\rm{int}}D)\cup_{\partial}M\ddot{o}$

$\bullet\bullet\bullet$

{xy}

(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