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In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham〔Fefferman, C. and Graham, R. "Conformal invariants", in ''Élie Cartan et les Mathématiques d'Aujourdui'', Asterisque (1985), 95-116.〕 for which a conformal manifold of dimension ''n'' is realized (''ambiently'') as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian manifold. The ambient construction is canonical in the sense that it is performed only using the conformal class of the metric: it is conformally invariant. However, the construction only works asymptotically, up to a certain order of approximation. There is, in general, an obstruction to continuing this extension past the critical order. The obstruction itself is of tensorial character, and is known as the (conformal) obstruction tensor. It is, along with the Weyl tensor, one of the two primitive invariants in conformal differential geometry. Aside from the obstruction tensor, the ambient construction can be used to define a class of conformally invariant differential operators known as the GJMS operators.〔Graham, R., Jenne, R., Mason, L.J., and Sparling, G.A.J. "Conformally invarant powers of the Laplacian I: Existence", ''Jour. Lond. Math. Soc'', 46 (1992), 557-565.〕 A related construction is the tractor bundle. ==Overview== The model flat geometry for the ambient construction is the future null cone in Minkowski space, with the origin deleted. The celestial sphere at infinity is the conformal manifold ''M'', and the null rays in the cone determine a line bundle over ''M''. Moreover, the null cone carries a metric which degenerates in the direction of the generators of the cone. The ambient construction in this flat model space then asks: if one is provided with such a line bundle, along with its degenerate metric, to what extent is it possible to ''extend'' the metric off the null cone in a canonical way, thus recovering the ambient Minkowski space? In formal terms, the degenerate metric supplies a Dirichlet boundary condition for the extension problem and, as it happens, the natural condition is for the extended metric to be Ricci flat (because of the normalization of the normal conformal connection.) The ambient construction generalizes this to the case when ''M'' is conformally curved, first by constructing a natural null line bundle ''N'' with a degenerate metric, and then solving the associated Dirichlet problem on ''N'' × (-1,1). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ambient construction」の詳細全文を読む スポンサード リンク
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