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665(2): １３２人目の素数さん [] 2023/03/22(水)16:07 ID:VqclUbtx(1/6)
>>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

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

668(1): １３２人目の素数さん [] 2023/03/22(水)16:10 ID:VqclUbtx(4/6)
>>667
つづき

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]).

Finally, according to Tanaka’s results, the choice
of the Cartan connection is controlled by the
∂-exact components of the curvature.

P24
Levi-Tanaka Algebra and Tanaka’s Prolongation Procedure
略
(引用終り)

669: １３２人目の素数さん [] 2023/03/22(水)16:55 ID:VqclUbtx(5/6)
>>668
>Levi-Tanaka Algebra and Tanaka’s Prolongation Procedure

下記か？
https://www.researchgate.net/publication/225316341_Classification_of_semisimple_Levi-Tanaka_algebras
Article PDF Available
Classification of semisimple Levi-Tanaka algebras
January 1998Annali di Matematica Pura ed Applicata 174(1):285-349
Authors:
Costantino Medori
Universita di Parma
Mauro Nacinovich
University of Rome Tor Vergata

https://en.wikipedia.org/wiki/L%C3%A9vy_process
Levy process
In probability theory, a Levy process, named after the French mathematician Paul Levy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Levy process may thus be viewed as the continuous-time analog of a random walk.

https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_(mathematician)
Paul Pierre Levy (15 September 1886 ? 15 December 1971)[2] was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions.

https://en.wikipedia.org/wiki/Tanaka%27s_formula
Tanaka's formula
https://en.wikipedia.org/wiki/Tanaka_equation
Tanaka equation

https://en.wikipedia.org/wiki/Local_time_(mathematics)
Local time (mathematics)
Local time appears in various stochastic integration formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields.
Tanaka's formula

671: １３２人目の素数さん [] 2023/03/22(水)17:34 ID:VqclUbtx(6/6)
>>670
＞LeviとLevyを混同しないでほしい

おっと
失礼しました
まさか、下記Tullio Levi-Civita （トゥーリオ・レヴィ＝チヴィタ）？
もしそうなら、全く不勉強でしたｍ（＿＿）ｍ

https://ja.wikipedia.org/wiki/%E3%83%88%E3%82%A5%E3%83%BC%E3%83%AA%E3%82%AA%E3%83%BB%E3%83%AC%E3%83%B4%E3%82%A3%EF%BC%9D%E3%83%81%E3%83%B4%E3%82%A3%E3%82%BF
トゥーリオ・レヴィ＝チヴィタ

トゥーリオ・レヴィ＝チヴィタ（Tullio Levi-Civita、1873年3月29日 - 1941年12月29日）は、イタリアのパドヴァ出身のユダヤ人数学者。テンソル解析学（絶対微分学）に貢献し、レヴィ＝チヴィタ記号（エディントンのイプシロン）の考案者として名高い。また、レヴィ・チヴィタ接続（en:Levi-Civita connection）やレヴィ＝チヴィタ (クレーター)（en:Levi-Civita (crater)）に名前が伝わっている。

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