In this article, we study convex affine domains which can cover a compact affine manifold.For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least C 1 boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional paraboloid is the only strictly convex quasi-homogeneous affine domain in R n up to affine equivalence. Furthermore we prove that if a strictly convex quasi-homogeneous projective domain is C α on an open subset of its boundary, then it is C α everywhere.Using this fact and the properties of asymptotic cones we find all possible shapes for developing images of compact convex affine manifolds with dimension ≤ 4.2000 Mathematics Subject Classification. 52A20, 57M50.

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In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

In this paper we show that any properly convex quasi-homogeneous affine domain with irreducible projective automorphism group is projectively equivalent to a homogeneous affine domain. Then as an application we answer positively the Markus conjecture when the manifold M is in a certain class of convex affine manifolds.Mathematics Subject Classification (1991): 51M10, 57S25

In this article, we investigate projective domains with a strictly convex point in the boundary and their automorphisms. We prove that ellipsoids can be characterized as follows: A domain Ω is an ellipsoid if and only if ∂Ω is locally strongly convex at some boundary point where an Aut(Ω)-orbit accumulates. We also show that every quasi-homogeneous projective domain in an affine space which is locally strictly convex at a boundary point, is the universal covering of a closed projective manifold.