Projective Connection - Projective Space As The Model Geometry

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). 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:

\left( \begin{matrix} \lambda&v^i\\ w_j&a_j^i \end{matrix} \right),\quad (v^i)\in {\mathbb R}^{1\times n}, (w_j)\in {\mathbb R}^{n\times 1}, (a_j^i)\in {\mathbb R}^{n\times n}, \lambda = -\sum_i a_i^i .

And consists of all these matrices with (wj) = 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

dζ + ∑i αi∧αi = 0
dαj + αj∧ζ + ∑k αjk∧αk = 0
dαji + αi∧αj + ∑k αki∧αjk = 0
dαi + ζ∧αi + ∑kαk∧αki = 0

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