##
*Reiko Miyaoka *

Hypersurface geometry and Hamiltonian systems

of hydrodynamic type

*(Lobachevskii Journal of Mathematics,
Vol.3, pp.209-220)*

Around 1982-1988, Dobrovin and Novikov gave a mathematical formulation
of Hamiltonian systems of hydrodynamic type, which appear in gas dynamics,
gas chromatography, etc. In 1991, Ferapontov showed one to one correspondence
between these systems and the hypersurfaces in the (pseudo-)Euclidean space.
Using this, he obtained the classification of the integrable non-diagonalizable
$3\times 3$ Hamiltonian systems of hydrodynamic type via the classification
of Dupin hypersurfaces in $E^4$ due to Pinkall. Furthermore, he showed
that the systems corresponding to the homogeneous isoparametric hypersurfaces
in $S^n$ with more than two principal curvatures are non-diagonalizable
but integrable, by deforming the systems to the n-wave systems which are
known to be integrable.

Isoparametric hypersurfaces with less than four principal curvatures
are known to be all homogeneous, while there exist infinitely many non-homogeneous
examples with four principal curvatures. S.T. Yau posed a question to classify
isoparametric hypersurfaces with four or six principal curvatures, which
is the remaining possible cases. Recently, the author proved that isoparametric
hypersurfaces with six principal curvatures are homogeneous. The idea of
the proof is a use of ``isospectral principle", which holds for a one-parameter
family of (self-adjoint) operators satisfying the Lax equation. This is
a basic principle to solve the Cauchy problem of the KdV equation, the
non-linear Schr\"odinger equation, the sine-Gordon equation and so forth,
via the inverse scattering method. Now, it turned out that all isoparametric
hypersurfaces other than those with four principal curvatures correspond
to the integrable systems.

Moreover, Ferapontov showed that the Lie transformation of hypersurfaces
corresponds to the ``reciprocal transformation" of the Hamiltonian systems,
which is a transformation preserving the diagonalizability, integrability
and linearly degeneracy of the isoparametric hypersurfaces under the Lie
transformations correspond also to the integrable systems. These hypersurfaces
belong to the hypersurface class called Dupin. Therefore the classification
of Dupin hypersurfaces is also interesting and important.

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