| A convex real projective structure on a smooth surface M is a representation of M as a quotient of a convex domain W⊂RP2 by a discrete group G⊂PGL&parl0;3,R&parr0; acting properly and freely on W . If c&parl0;M&parr0;<0, then the equivalence classes of convex real projective structures form a moduli space B&parl0;M&parr0; which is an extension of the Teichmuller space T&parl0;M&parr0; .;In Chapter 1, we study the set of positive hyperbolic elements Hyp+ ⊂PGL&parl0;3,R&parr0; since the holonomy group G of a convex real projective structure lies in Hyp +.;In Chapter 2, we study Hom(pi,G)/G for pi = pi1(M) and an algebraic group G since B&parl0;M&parr0; embeds onto an open subset of Hom(pi, PGL(3, R ))/PGL (3, R ). We induce a symplectic form o on B&parl0;S&parr0; for a closed Riemann surface S by Fox's calculus and the fundamental cycle.;In Chapter 3, we extend theories B&parl0;M&parr0; of for a compact oriented Riemann surface M with boundary. Since B&parl0;M&parr0; is a Poisson manifold, we find a symplectic foliation called the parabolic foliation by giving a restriction on the boundary components. We show the modified form w&d5; is a symplectic form on each parabolic leaf.;In Chapter 4, we show (S, J, g, o) is an almost Kahler manifold where S is a closed Riemann surface, J is an almost complex structure, g is the Weil-Petersson metric and o is symplectic defined in Chapter 2 and study the integrability condition.;In Appendix, we actually calculate the symplectic forms w&d5; for the pair of pants and the functured torus cases with using Mathematica. |
