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

Read more about this topic:  Projective Connection

Famous quotes containing the words space, model and/or geometry:

    “Play is a major avenue for learning to manage anxiety. It gives the child a safe space where she can experiment at will, suspending the rules and constraints of physical and social reality. In play, the child becomes master rather than subject.... Play allows the child to transcend passivity and to become the active doer of what happens around her.”
    —Alicia F. Lieberman (20th century)

    “I’d like to be the first model who becomes a woman.”
    —Lauren Hutton (b. 1944)

    “I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.”
    —Ralph Waldo Emerson (1803–1882)