[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ2 (1002ﾚｽ)
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666(1): １３２人目の素数さん [] 2023/03/22(水)16:08 ID:VqclUbtx(2/6)
>>665
>>660
>田中昇とElie Cartanを知らない人のために
ありがとう
他意はないが、抜粋貼る
（こうしておけば、一般検索から、ここに到達する人がいるので）
https://www.ams.org/notices/201101/rtx110100020p.pdf
Notices of the AMS Volume 58, Number 1 January 2011
From Cartan to Tanaka:Getting Real in the Complex World
Vladimir Ezhov, Ben McLaughlin, and Gerd Schmalz
It is well known from undergraduate complex analysis that holomorphic functions of one complex variable are fully determined by their values at the boundary of a complex domain via the Cauchy integral formula. This is the first instance in which students encounter the general principle of complex analysis in one and several variables that the study of holomorphic objects often reduces to the study of their boundary values. The boundaries of complex domains, having odd topological dimension, cannot be complex objects. This motivated the study of the geometry of real hypersurfaces in complex space. In particular, since all established facts about a particular hypersurface carry over to its image via a biholomorphic mapping in the ambient space, it is important to decide which hypersurfaces are equivalent with respect to such mappings - that is, to solve an equivalence problem for real hypersurfaces in a complex space.
つづく
667(1): １３２人目の素数さん [] 2023/03/22(水)16:10 ID:VqclUbtx(3/6)
>>666 被った、貼り直し
>>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
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