Vector fields on nonorientable surfaces

A one-to-one correspondence is established between the germs of functions and tangent vectors on a NOS X and the bi-germs of functions, respectively, elementary fields of tangent vectors (EFTV) on the orientable double cover of X. Some representation theorems for the algebra of germs of functions, the tangent space at an arbitrary point of X, and the space of vector fields on X are proved by using a symmetrisation process. An example related to the normal derivative on the border of the Möbius strip supports the nontriviality of the concepts introduced in this paper.