xii A BRIEF HISTORY
N = max{sn + 2n, sn + n -f 5}. It is still not clear whether this is the best possible
result on the dimension of the ambient space.
In 1970, Gromov and Rokhlin and Greene, independently, proved that any n-
dimensional smooth Riemannian manifold admits a smooth isometric embedding
in Rs™+n locally. The proof is based on the iteration scheme introduced by Nash.
For 2-dimensional Riemannian manifolds, better results are available. First,
according to Gromov or Giinther, any compact 2-dimensional smooth Riemannian
manifold can be isometrically embedded in R10 smoothly. When the manifolds have
some special property, the dimension of the ambient space can be lowered. On the
other hand, in 1973 Poznyak proved that any smooth 2-dimensional Riemannian
manifold can be locally isometrically embedded in R4 smoothly. We are more
interested in the question of whether we can isometrically embed a 2-dimensional
Riemannian manifold in R3, locally or globally.
2. Local Isometric Embedding of Surfaces in
R3.
It was known to Darboux in 1894 that isometrically embedding a surface locally
in
R3
is equivalent to finding a local solution of some nonlinear equation of the
Monge-Ampere type. Such an equation is now called the Darboux equation; its type
is determined by the sign of the Gauss curvature K. It is elliptic if K is positive and
hyperbolic if K is negative. It is degenerate if K vanishes. Remarkably, even today,
the local solvability of the Darboux equation in the general case is not covered by
any known theory of partial differential equations.
The first attempt to establish the local isometric embedding of surfaces in R3
was not through the Darboux equation. In 1908, Levi proved the local isometric
embedding in
R3
of surfaces with negative curvature by using the equations of vir-
tual asymptotes. It was several decades later that the Darboux equation attracted
the attention of those interested in the isometric embedding. In the early 1950's,
Hartman and Winter studied the Darboux equation in the case when the Gauss
curvature K does not vanish and proved the existence of local solutions to the
Darboux equation and hence the local isometric embedding in
R3
in that case.
For a long time the case when K vanishes did not give way to the efforts of
mathematicians. In 1985 and 1986, Lin made important breakthroughs. By a
delicate analysis, he obtained the existence of sufficiently smooth local solutions
of the Darboux equation and hence a sufficiently smooth isometric embedding in
a neighborhood of p for the following two cases: K(p) = 0 and dK(p) ^ 0, or
K 0 in a neighborhood of the point p. Later, in 1987, Nakamura proved the
existence of the smooth local isometric embedding if K(p) = 0, dK{p) 0 and
HessK(p) 0. Evidently, K is nonpositive near the point p and the leading part
of K is an irreducible quadratic polynomial. For the case of nonpositive Gauss
curvature, Hong in 1991 also proved the existence of a sufficiently smooth local
isometric embedding in a neighborhood of p if K hg2rn, where h is a negative
function and g is a function with g(p) = 0 and dg(p) ^ 0. In 2005, Han gave
a simple proof of Lin's result that g admits a sufficiently smooth local isometric
embedding in a neighborhood of p if K(p) = 0 and dK(p) ^ 0. All these results
are based on a careful study of the Darboux equation.
In 2003, Han, Hong and Lin studied the local isometric embedding of surfaces
in R3 by a different method. Instead of the Darboux equation, they studied a
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