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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