Abstract
This chapter discusses a nonlinear problem in differential geometry. Any metric ds2 on a two-dimensional sphere S2determines a Gauss curvature function K satisfying the Gauss–Bonnet formula. This chapter discusses the converse question: to characterize all smooth functions on a two-dimensional sphere which can be obtained in this manner from some riemannian metric. To characterize all Gauss curvature functions belonging to metrics ds2 which are conformally related to the standard metric ds02, so that ds2 = λds02, where λ is a positive function on the sphere. This requires the determination of the single function λ in terms of the given function K.
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