XIV

A BRIEF HISTORY

no more than two times continuously differentiable metrics. For compact surfaces

with Gauss curvature of mixed sign, Alexandrov in 1938 introduced a class of sur-

faces satisfying some integral condition for its Gauss curvature and proved that any

compact analytic surface with this condition is rigid. In 1963, Nirenberg general-

ized this result to smooth surfaces. To do this, he needed some extra conditions,

one of which is not intrinsic.

4. Local Isometric Embedding of n-Dimensional Riemannian Manifolds in RSn.

For the local isometric embedding of smooth n-dimensional Riemannian man-

ifolds in

RSri,

the case n 3 is sharply different from the case n = 2. For n = 2,

there is only one curvature function, and it determines the type of the Darboux

equation, that is, the equation for the isometric embedding of 2-dimensional Rie-

mannian manifolds in

R3.

For n 3, the role of curvature functions is not clear.

In 1983, Bryant, Griffiths and Yang studied the characteristic varieties associ-

ated with the differential systems for the isometric embedding in

RSn

of smooth n-

dimensional Riemannian manifolds. They proved that these characteristic varieties

are never empty if n 3. This implies in particular that the differential systems for

the isometric embedding in Rs™ of n-dimensional Riemannian manifolds are never

elliptic for n 3, no matter what assumptions are put on curvatures. This is a

sharp difference from the case n = 2. A related result is the local rigidity proved

by Berger, Bryant and Griffiths in 1983.

For n = 3, Bryant, Griffiths and Yang in 1983 studied the characteristic va-

rieties in detail. They were able to classify the type of differential system for the

isometric embedding by its curvature functions. Here an important quantity is

the signature of the curvature tensor viewed as a symmetric linear operator acting

on the space of 2-forms. They proved that any smooth 3-dimensional Riemann-

ian manifold admits a smooth local isometric embedding in

R6

if the signature is

different from (0,0) and (0,1). Then in 1989, Nakamura and Maeda proved the

existence of the smooth local isometric embedding in

R6

of smooth 3-dimensional

Riemannian manifolds if the curvature tensors are not zero. The key step in the

proof is the local existence of solutions to differential systems of principal type.