PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 63 (77), pp. 4754 (1998) 

Some remarks on the nonorientable surfacesI. Bârza, D. Ghisa, S. IanusDepartment of Mathematics, Lulea University of Technology, Lulea, SwedenAbstract: 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 (MSC2000): 30F50 Full text of the article:
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