View source for One-sided and two-sided surfaces

Two types of surfaces, differing in the way in which they are situated in the ambient space (one-sided position and two-sided position). For example, the cylinder is a two-sided surface, while the [[Möbius strip|Möbius strip]] is a one-sided surface. A characteristic distinction between these surfaces is that the boundary of the cylinder consists of two curves, while the boundary of the Möbius strip is a single curve. Among the closed surfaces the [[Sphere|sphere]] and the [[Torus|torus]] are two-sided, while the [[Klein surface|Klein surface]] is one-sided. As examples of two-sided and one-sided situations one may cite imbeddings of the circle in the Möbius strip. Thus, the cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682301.png" /> (see Fig.) is a one-sided curve, while the cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682302.png" /> is two-sided (in general, any [[Desorienting path|desorienting path]] lies in the surface one-sidedly).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o068230a.gif" />

Figure: o068230a

More precisely, one- and two-sided surfaces are two types of manifolds differing in the way in which they are imbedded in the ambient space (of dimension higher by one). Two-sidedness and one-sidedness are related to orientability and non-orientability (see [[Orientation|Orientation]]), but unlike these, are not intrinsic properties of the surface and depend on the ambient space. For example, there exist orientable two-sided surfaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682304.png" />; non-orientable two-sided surfaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682305.png" />; orientable one-sided surfaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682306.png" />; non-orientable one-sided surfaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682307.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682308.png" /> is the sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o0682309.png" /> the torus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823010.png" /> the projective plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823011.png" /> projective space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823012.png" /> a desorienting path on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823013.png" />).

In an orientable space (e.g. in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823014.png" />) a hypersurface is orientable if and only if it is two-sided.

Suppose that a normal vector is moved along a closed curve on a smooth surface immersed in some space in such a way that it remains normal. If upon return to the starting point the direction of the normal is the same as it was originally, independently of the choice of curve, then the surface is called two-sided; in the opposite case it is called one-sided. More generally, a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823015.png" /> is situated two-sidedly if and only if its [[Normal bundle|normal bundle]] is trivial (there exists a non-zero section in this bundle). Conversely, the normal bundle of a one-sided surface is non-trivial: There exists a curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823016.png" /> over which the bundle is a Möbius strip.

Locally every (hyper) surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823017.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823018.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823019.png" /> into two parts, that is, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823020.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823022.png" /> consists of two components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823025.png" /> belongs to their common boundary. On the other hand, a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823027.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823028.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823029.png" />) has either one or two components the boundaries of which contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823030.png" />. In the first case, the (hyper) surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823031.png" /> is also called one-sided, and in the second case, two-sided. Thus, although the surface locally has two sides, globally it may be one-sided, while, in contrast, a two-sided surface need not divide its neighbourhood in the space.

For a two-sided surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823032.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823033.png" />, the [[Intersection index (in homology)|intersection index (in homology)]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823034.png" /> of any closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823036.png" /> satisfies the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823037.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823038.png" />. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823039.png" /> is one-sided, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823040.png" /> for some curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068230/o06823041.png" />. This fact can also be taken as the definition of one- and two-sidedness (along with the movement of a normal vector and the division of a neighbourhood).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   S.E. Cohn-Vossen,   "Geometry and the imagination" , Chelsea  (1952)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Seifert,   W. Threlfall,   "A textbook of topology" , Acad. Press  (1980)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.B. Fuks,   A.T. Fomenko,   V.L. Gutenmakher,   "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR></table>