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Such embeddings are shown to exist without exception for m è 2, but with certain necessary exceptions for rn = l, the result in this case standing as a converse to the classical Four Vertex Theorem for plane curves. While the above problem lies in the realm of differential geometry in the large, its solution comes via differential topology by studying deformations of normal vector fields on a closed smooth manifold M in R. Detailed proofs may be found in [9], while a leisurely exposition of the converse to the Four Vertex Theorem appears in [lO]. I am indebted to Eugenio Calabi, Jerry Kazdan and Frank Warner for many helpful conversations, and particularly to Warner for introducing me to the problem in the first place and for pointing out its relation to the study of normal vector fields on spheres.

Deformations of normal vector fields and the generalized Minkowski problem