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On some applications of Cartan's generalization of Lie's Third Fundamental Theorem (Lie Groups, Geometric Structures and Differential Equations : One Hundred Years after Sophus Lie)

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On

some applications of Cartan’s generalization of Lie’s Third

Fundamental Theorem

Robert Bryant (Duke Univ.)

Lie’s Third Fundamental Theorem asscrts the cxistencc of a Lic algebra wh$oscsCructurc$

constants are given, subject to the condition that the constants satisfythe Jacobi $e(1^{u_{\dot{c}}\iota\downarrow,ion}\cdot$

Cartan

generalized this theorem to an existence theorcm for coframings whosr structure

functions (nolongerconstants) satisfycertain systems ofpartial$diH^{\cdot}erential$ equ$\dot{‘}\iota Ciol\iota s$. $\prime rhis$

result has not had the wide application that

one

might have expected, possibly $bec_{\dot{c}}\iota use$

the result is not well known outside of the theory of exterior differential systeIns. In this

lecture, I describe some applications of this theorem to problems in general $re1_{c}^{C}\iota ti\backslash \prime ity.$

,

CR

geometry, prescribed holonomy and curvature problems, and

so on.

数理解析研究所講究録

参照

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