23/03/22 16:10:00.77 VqclUbtx.net
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>>665
つづき
In the case of one complex variable, the Riemann
mapping theorem says that any simply connected
domain is either C or equivalent to the unit disc. In
contrast, Henri Poincare [17] showed that in higher
dimensions even the ball and the bidisc are not
equivalent, which implies that their boundaries
cannot be equivalent.
In the same article Poincare posed the local
equivalence problem, i.e., to decide when two hypersurfaces are equivalent in the neighbourhoods
of given points. He sketched a heuristic argument
that any two real hypersurfaces in C2 cannot be
expected to be locally equivalent.
In order to solve this equivalence problem
for real hypersurfaces in C2, Elie Cartan [6], [7]
constructed in 1932 a “hyperspherical connection” by applying his method of moving frames.
The technique of Cartan has been further developed by introducing modern geometric and
algebraic tools, mainly in the groundbreaking
work by Noboru Tanaka (see [22], [23], [24]).
These powerful and elegant methods are widely
used in conformal geometry and have led to the
development of parabolic geometry (see [5]), while
Cartan’s original approach, applied to hypersurfaces in higher dimensional complex space by
Shiing-Shen Chern and Jurgen Moser [8], is still
dominant in complex analysis (see, e.g., [12], [13]).
つづく