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In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of ''p''-forms. The case ''p'' = 1 was introduced by Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges Poisson geometry and presymplectic geometry, based on work with his advisor Alan Weinstein. The twisted version of the Courant bracket was introduced in 2001 by Pavol Severa, and studied in collaboration with Weinstein. Today a complex version of the ''p''=1 Courant bracket plays a central role in the field of generalized complex geometry, introduced by Nigel Hitchin in 2002. Closure under the Courant bracket is the integrability condition of a generalized almost complex structure. ==Definition== Let ''X'' and ''Y'' be vector fields on an N-dimensional real manifold ''M'' and let ''ξ'' and ''η'' be ''p''-forms. Then ''X+ξ'' and ''Y+η'' are sections of the direct sum of the tangent bundle and the bundle of ''p''-forms. The Courant bracket of ''X+ξ'' and ''Y+η'' is defined to be : where is the Lie derivative along the vector field ''X'', ''d'' is the exterior derivative and ''i'' is the interior product. 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「Courant bracket」の詳細全文を読む スポンサード リンク
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