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Concept
Projective connection
Résumé
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, 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.
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 of the homogeneous space is the projective space RPn which we shall represent by homogeneous coordinates . The symmetry group of is G = PSL(n+1,R). Let H be the isotropy group of the point . Thus, M = G/H presents 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 matrices:
And consists of all these matrices with . Relative to the matrix representation above, the Maurer-Cartan form of G is a system of 1-forms satisfying the structural equations (written using the Einstein summation convention):
A projective structure is a linear geometry on a manifold in which two nearby points are connected by a line (i.e., an unparametrized geodesic) in a unique manner. Furthermore, an infinitesimal neighborhood of each point is equipped with a class of projective frames. According to Cartan (1924),
Une variété (ou espace) à connexion projective est une variété numérique qui, au voisinage immédiat de chaque point, présente tous les caractères d'un espace projectif et douée de plus d'une loi permettant de raccorder en un seul espace projectif les deux petits morceaux qui entourent deux points infiniment voisins.
Source officielle
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