The properties of uniformly rotating white dwarfs (RWDs) are analyzed within the framework of general relativity. Hartle's formalism is applied to construct the internal and external solutions to the Einstein equations. The white dwarf (WD) matter is described by the relativistic Feynman-Metropolis-Teller equation of state which generalizes that of Salpeter by taking into account the finite size of the nuclei, and the Coulomb interactions as well as electroweak equilibrium in a self-consistent relativistic fashion. The mass M, radius R, angular momentum J, eccentricity , and quadrupole moment Q of RWDs are calculated as a function of the central density ρ c and rotation angular velocity Ω. We construct the region of stability of RWDs (J-M plane) taking into account the mass-shedding limit, inverse β-decay instability, and the boundary established by the turning points of constant J sequences which separates stable from secularly unstable configurations. We found the minimum rotation periods ~0.3, 0.5, 0.7, and 2.2 s and maximum masses ~1.500, 1.474, 1.467, 1.202 M ☉ for 4He, 12C, 16O, and 56Fe WDs, respectively. By using the turning-point method, we found that RWDs can indeed be axisymmetrically unstable and we give the range of WD parameters where this occurs. We also construct constant rest-mass evolution tracks of RWDs at fixed chemical composition and show that, by losing angular momentum, sub-Chandrasekhar RWDs (mass smaller than maximum static one) can experience both spin-up and spin-down epochs depending on their initial mass and rotation period, while super-Chandrasekhar RWDs (mass larger than maximum static one) only spin up.
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