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:

    “Though seas and land be ‘twixt us both,
    Our faith and troth,
    Like separated souls,
    All time and space controls:
    Above the highest sphere we meet
    Unseen, unknown, and greet as angels greet.”
    —Richard Lovelace (1618–1658)

    “One of the most important things we adults can do for young children is to model the kind of person we would like them to be.”
    —Carol B. Hillman (20th century)

    “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)