Geodesic
Some Remarks on the Nonorientable Surfaces
Publications de l'Institut Mathématique, _N_S_63 (1998) no. 77, p. 47 .

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It is a classical result of F. Klein that for any nonorientable (regular enough) surface $\boldkey X $ there is an orientable surface ${\Cal O}_2$ and an involution without fixed point of ${\Cal O}_2$ such that $\boldkey X $ is isomorphic to the quotient space of ${\Cal O}_2$ with respect to the group generated by the respective involution. In this note a reinforcement of the Klein's result is presented and the effect on the vector bundle of covariant tensors of second order on X produced by that involution is studied.The projection $p:{\Cal O}_2 \longto \boldkey X $ induces an isomorphism between the vector space of covariant tensors of order two on $\boldkey X$ and the space of covariant symmetric tensors of order two on ${\Cal O}_2$ which are invariant with respect to the given involution.
Classification : 30F50 
@article{PIM_1998_N_S_63_77_a6,
     author = {I. Barza and D. Ghisa and Stere Ianus},
     title = {Some {Remarks} on the {Nonorientable} {Surfaces}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {47 },
     publisher = {mathdoc},
     volume = {_N_S_63},
     number = {77},
     year = {1998},
     zbl = {1002.53007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1998_N_S_63_77_a6/}
}
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I. Barza; D. Ghisa; Stere Ianus. Some Remarks on the Nonorientable Surfaces. Publications de l'Institut Mathématique, _N_S_63 (1998) no. 77, p. 47 . http://geodesic.mathdoc.fr/item/PIM_1998_N_S_63_77_a6/