In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant *negative* Gaussian curvature K immersed in $ R^3 $. This theorem answers the question for the negative case of which surfaces in $ R^3 $ can be obtained by isometrically immersing complete manifolds with constant curvature.(Wikipedia)

Among constant *positive* Gaussian curvature K surfaces immersed in $ R^3 $ are included the hyperbolic types (cheese tire with inward cuspidal edges), incomplete and not regular.

Let us hypothetically say someone attempts to prove irregularity of some of these positive surfaces as well. Which Lemmas should be included in proof of contradiction for cheese tires distinguishing or setting them apart from from the other regular surfaces, viz. spindles and Riemann spheres?