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:

    “And Space with gaunt grey eyes and her brother Time
    Wheeling and whispering come,”
    —James Elroy Flecker (1884–1919)

    “When Titian was mixing brown madder,
    His model was posed up a ladder.
    Said Titian, “That position
    Calls for coition,”
    So he lept up the ladder and had her.”
    —Anonymous.

    “... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
    —Louise Bourgeois (b. 1911)