Projective connection

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 M of the homogeneous space is the projective space RPⁿ which we shall represent by homogeneous coordinates [x₀,...,x_n]. The symmetry group of M is G = PSL(n+1,R). Let H be the isotropy group of the point [1,0,0,...,0]. Thus, M = G/H presents M as a homogeneous space.

Let ${\displaystyle {\mathfrak {g}}}$ be the Lie algebra of G, and ${\displaystyle {\mathfrak {h}}}$ that of H. Note that ${\displaystyle {\mathfrak {g}}={\mathfrak {s}}{\mathfrak {l}}(n+1,{\mathbb {R} })}$. As matrices relative to the homogeneous basis, ${\displaystyle {\mathfrak {g}}}$ consists of trace-free (n+1)×(n+1) matrices:

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