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Real projective structures on $n$-orbifolds are useful in understanding thespace of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolicmanifolds deform to manifolds with such structures not projectively equivalentto the original ones. The purpose of this paper is to understand the structuresof properly convex ends of real projective $n$-dimensional orbifolds. Inparticular, these have the radial or totally geodesic ends. For this, we willstudy the natural conditions on eigenvalues of holonomy representations of endswhen these ends are manageably understandable. In this paper, we only study theproperly convex ends. The main techniques are the Vinberg duality and ageneralization of the work of Goldman, Labourie, and Margulis on flatLorentzian $3$-manifolds. Finally, we show that a noncompact strongly tameproperly convex real projective orbifold with generalized lens-type orhorospherical ends satisfying some topological conditions always has a stronglyirreducible holonomy group.

The classification of ends of properly convex real projective orbifolds II: Properly convex radial ends and totally geodesic ends