Conformal group and fundamental theorem for a class of symmetric spaces

Abstract

Using a geometric approach, we define and investigate the conformal group of a symmetric space with twist. In the non-degenerate case we characterize this group by a theorem generalizing the fundamental theorem of projective geometry. IntroductionThe real projective space RP n = O (n + 1)/( O (n) × O (1)) and the Sphere S n = SO(n + 1)/ SO(n) are examples of a class of symmetric spaces M = G/H which can also be written as a quotient M = G b /Q with respect to a "big Lie group" G b . In the first case, G b = P Gl(n + 1, R) is the general projective group, characterized by the fundamental theorem of projective geometry as the group preserving collinearity (if n > 1), and in the second case, G b = SO(n+1, 1) is the conformal group, characterized by a classical result of Liouville (1850) as the group preserving the Riemannian metric of the sphere up to a scalar function (if n > 2). Another example is the unitary group U (n) which is homogeneous under the "big group" SU(n, n). Here I.E. Segal offered the conjecture that the "big group" can be characterized as the causal group of U (n) ([Se76, p.35]).In this work we present a unified theory of the "big group". We consider not only symmetric spaces G/H which can globally be written as a quotient G b /Q, but also spaces having "locally" such a realization; in the semisimple case these are the "open symmetric orbits in symmetric R-spaces" classified by B.O. Makarevič ([Ma73]). In our previous work [Be97] we have shown that algebraically such spaces, called symmetric spaces with twist, are equivalent to Jordan triple systems (JTS), and we have given an intrinsic 40 W. Bertram geometric characterization which is the starting point for the presentation given here.Every symmetric space with twist admits a twisted complexicification; for example, if M = RP n , then the twisted complexification is the complex projective space CP n . In the present work we characterize the twisted complex spaces as circled spaces; these are complex manifolds with a real affine connection such that multiplication by i in each tangent space extends to a holomorphic affine (local) diffeomorphism (Chapter 1). By a sort of analytic continuation, we deduce that multiplication by real scalars in each tangent space also has a canonical extension to a local diffeomorphism (no longer affine). The corresponding local one-parameter group attached to each point p in the manifold induces a vector field E = E p which we call an Euler operator on a symmetric space. In the example of M = RP n , using the usual affinization V = R n of M , the Euler operator E = E 0 is nothing but the usual Euler operator E(x) = x on R n . In fact, to any Euler operator E p on a symmetric space we canonically associate a chart, called Jordan coordinates, such that E p is represented by the usual Euler operator (Chapter 2). Now, if M = G/H, then the Lie algebra g of G together with the Euler operators generate a "big Lie algebra" g b of vector fields on M , called the (inner) conformal Lie algebra. It is worth noticing that, becaus...