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In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold. The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections, though, projective connections also define geodesics. However, these geodesics are not affinely parametrized. Rather they are projectively parametrized, meaning that their preferred class of parameterizations is acted upon by the group of fractional linear transformations. Like an affine connection, projective connections have associated torsion and curvature. ==Projective space as the model geometry== The first step in defining any Cartan connection is to consider the flat case: in which the connection corresponds to the Maurer-Cartan form on a homogeneous space. In the projective setting, the underlying manifold ''M'' of the homogeneous space is the projective space RPn which we shall represent by homogeneous coordinates (). The symmetry group of ''M'' is ''G'' = PSL(''n''+1,R).〔It is also possible to use PGL(''n''+1,R), but PSL(''n''+1,R) is more convenient because it is connected.〕 Let ''H'' be the isotropy group of the point (). Thus, ''M'' = ''G''/''H'' presents ''M'' as a homogeneous space. Let be the Lie algebra of ''G'', and that of ''H''. Note that . As matrices relative to the homogeneous basis, consists of trace-free (''n''+1)×(''n''+1) matrices: :. And consists of all these matrices with (''w''j) = 0. Relative to the matrix representation above, the Maurer-Cartan form of ''G'' is a system of 1-forms (ζ, αj, αji, αi) satisfying the structural equations〔Cartan's approach was to derive the structural equations from the volume-preserving condition on ''SL''(''n''+1) so that explicit reference to the Lie algebra was not required.〕 :''d''ζ + ∑i αi∧αi = 0 :''d''αj + αj∧ζ + ∑k αjk∧αk = 0 :''d''αji + αi∧αj + ∑k αki∧αjk = 0 :''d''αi + ζ∧αi + ∑kαk∧αki = 0〔A point of interest is this last equation is completely integrable, which means that the fibres of ''G'' → ''G''/''H'' can be defined using only the Maurer-Cartan form, by the Frobenius integration theorem.〕 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「projective connection」の詳細全文を読む スポンサード リンク
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