General property of neutrino mass matrix and CP-violation

It is found that the atmospheric neutrino mixing angle of θatm is determined to be tanθatm=Im(B)/Im(C) for B=Mνeνμ and C=Mνeντ, where Mij is the ij element of Mν†Mν with Mν as a complex symmetric neutrino mass matrix in the (νe, νμ, ντ)-basis. Another mixing angle, θ13, defined as Ue3=sinθ13e−iδ is subject to the condition: tan2θ13∝|sinθatmB+cosθatmC| and the CP-violating Dirac phase of δ is identical to the phase of sinθatmB*+cosθatmC*. The smallest value of |sinθ13| is achieved at tanθatm=−Re(C)/Re(B) that yields the maximal CP-violation and that implies C=−κB* for the maximal atmospheric neutrino mixing of tanθatm=κ=±1. The generic smallness of |sinθ13| can be ascribed to the tiny violation of the electron number conservation.