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

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

$\\bullet\\bullet\\bullet$

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