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.
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