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