A Brief History Differential geometry began as the study of curves and surfaces in 3-space. The concept of a Riemannian manifold, an abstract manifold with a metric structure, was first formulated by Riemann in 1868 to generalize these classical objects. Natu- rally there arose the question of whether an abstract Riemannian manifold is simply a submanifold of some Euclidean space with its induced metric. This is the iso- metric embedding question. It has assumed a position of fundamental conceptual importance in differential geometry. We briefly review four important aspects of the field of isometric embeddings of Riemannian manifolds in Euclidean space. 1. General Isometric Embedding of Riemannian Manifolds. In 1873, Schlaefli made the following conjecture: Every n-dimensional smooth Riemannian manifold admits a smooth local isometric embedding in RSn, with sn n(n + l)/2. It was more than 50 years later that an affirmative answer was given for the analytic case successively by Janet and Cartan they proved in 1926-1927 that any analytic n-dimensional Riemannian manifold has a local analytic isometric embedding in RSn. Schlaefli's question for the smooth case when n = 2 was given renewed attention by Yau in the 1980's and 1990's. For the global isometric embedding, Nash in 1954 and Kuiper in 1955 proved the existence of a global C1 isometric embedding of n-dimensional Riemannian manifolds in M2n+1. For smooth isometric embeddings, the difficulty arises from the loss of derivatives in the attempt to solve the nonlinear equations corresponding to the isometric embedding. In an outstanding paper published in 1956, Nash introduced an important technique of using smoothing operators to make up for the loss of derivatives. He proved that any smooth n-dimensional Riemannian manifold admits a (global) smooth isometric embedding in the Euclidean space RN, for N = 3sn + 4n in the compact case and N = (n + l)(3sn + 4n) in the noncompact case. The technique proves to be extremely useful in solving nonlinear differential equations. It has been modified by many people, including Moser and Hormander, and is now known as the hard implicit function theorem, or Nash-Moser iteration. Following Nash, one naturally looks for the smallest N. In his book Partial Differential Relations, published in 1986, Gromov studied various problems related to the isometric embedding of Riemannian manifolds. He proved that N = sn + 2n + 3 is enough for the compact case. Then in 1989, Giinther vastly simplified Nash's original proof. By rewriting the differential equations cleverly, he was able to employ the contraction mapping principle, instead of the Nash-Moser iteration, to construct solutions. Giinther also improved the dimension of the target space to xi
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