Shiing-Shen Chern, Nicolaas H. Kuiper, Some Theorems on the Isometric Imbedding of Compact Riemann Manifolds in Euclidean Space, Annals of Mathematics, Second Series, Vol. 56, No. 3 (Nov., 1952), pp. 422-430
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Chern et al. (1952) studied this question.