Elastic scattering of transversely polarized electron antineutrino beam off unpolarized electrons can be used to detect the CP-violating effects by measuring the azimuthal asymmetry of recoil electrons caused by the interference terms between the standard vector $c_{V}^{L}$, axial $c_{A}^{L}$ couplings of left-chirality antineutrinos and exotic scalar $c_{S}^{R}$ coupling of right-chirality ones in the differential cross section. It would be a positive evidence for the existence of the exotic antineutrino states. Moreover, we also show that the differential cross section for the $overline{ u} e^- $ scattering can be obtained from the one for the $ u e$ scattering, if one makes the substitution $c_{T}^{R} to -c_{T}^{R}$, $c_{A}^{L} to -c_{A}^{L}$, ${bf q}to -{bf q}$, ${boldmath $hat{eta}_{ u}$}to -{boldmath $hat{eta}_{ u}$}$.
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