On the basis of the tight-binding theory, we derive the band structure formula of the deformed carbon nanotube. The results show that the band gap varies with increasing deformation, which shows that the deformation can change the nature of the conductivity of carbon between the metal-type and semiconductor-type. Especially for zigzag nanotubes，there are three different rules goveoning this translation, which is dependent on the remainder of n divided by 3. Furthermore, by using Natori theory on the field-effect transistor we study the current-voltage characteristics of deformed carbon nanotube field-effect transistors. For strain deformation, the conductivity of zigzag nanotube presents different characteristics with the remainder of n by 3. For armchair nanotubes, the conductivity does not change with the strain parameter. However, for torsion deformation, the conductivity rapidly increases, specially for the armchair tube. It has many obviously different conductivity behaviors between zigzag tubes and armchair tubes. At some special torsion angles, the conductivity of zigzag tube and armchair tube has obvious changes, which reveals the translation between metal-type and semiconductor-type conductivities.