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I have read about the orientation of manifold in the Tu's book. The book is very readable but the first example about non-orientable manifold is seemly hard to understand. On page 208, he gave an argument that Möbius band is not orientable. Can anybody help me by some instructions?

Thanks.

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  • $\begingroup$ One question mark is enough to indicate a question. Also, titles should contain somewhat coherent text, not disjoint keywords. $\endgroup$
    – user147263
    Aug 14 '14 at 20:09
  • $\begingroup$ Which book are you referring to? "An introduction to manifolds"? By Loring Tu? I ask because the discussion of the Mobius band is on page 241, not page 208... $\endgroup$
    – Sam Nead
    Aug 14 '14 at 20:19
  • $\begingroup$ Yes, The book is "An introduction to manifolds". I think the difference due to book's version. I read some proofs about non-orientatable of Mobius surface, such as Carmo book, which is easy to understand. $\endgroup$
    – user105132
    Aug 15 '14 at 14:18
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Try reading some other discussions (http://en.wikipedia.org/wiki/Orientability, http://en.wikipedia.org/wiki/M%C3%B6bius_strip) and then return to the book you are reading. Making a physical model is also a very good idea. Or just look at a cool picture:

http://alpha.zimage.com/~ant/antfarm/ants/moebius-ants.jpg

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Make a Mobius band from a strip of paper: glue the ends together with a half-twist.

Try to color one side red and the "other side" blue with crayons. You will find that you cannot distinguish one side from the other globally.

Seriously.

If you haven't tried this as a kid, you really should do this physical exercise now, it won't take long.

The Mobius strip is such a useful touchstone example it is worth it to understand it very well.

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