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A real projective structure on a 3-orbifold is given by locally modeling the orbifold by real projective geometry. We present some methodology to study Coxeter groups which are fundamental groups of 3-orbifolds with representations in and deformation spaces. There are related examples by Benoist. These examples give us nontrivial deformation spaces of projective structures. [ABSTRACT FROM AUTHOR]

The Lipschitz regularity is perhaps the most natural, and surely the most geometrical among all the types of regularities. For example, the Lipschitz character of an ordinary differential equation (vector field) is the natural classical sufficient condition for the (unique) integrability of this equation. The goal here is to show that, in some sense, the Lipschitz regularity is also necessary, if one assumes (geometric) individual conditions on the trajectories. In other words, we show that tangential rigidity leads to a transversal regularity. [ABSTRACT FROM AUTHOR]

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let G be a Lie group acting on a space X. We show that the space of isotopy-equivalence classes of (G, X)-structures on a compact orbifold Σ is locally homeomorphic to the space of representations of the orbifold fundamental group of Σ to G following the work of Thurston, Morgan, and Lok. This implies that the deformation space of (G, X)-structures on Σ is locally homeomorphic to the character variety of representations of the orbifold fundamental group to G when restricted to the region of proper conjugation action by G. [ABSTRACT FROM AUTHOR]

It is demonstrated that hypersurfaces Mn?An+1 with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory. In the case of surfaces M2?A3 this system reduces to a single third-order PDE, fxxxfyyy-fxxyfxyy=1 where x and y are the asymptotic coordinates on M2. [ABSTRACT FROM AUTHOR]

An affine manifold is a manifold with a flat affine structure, i.e. a torsion-free flat affine connection. We slightly generalize the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is zero confirming an old conjecture of Chern. The technique is from Kim and Lee's work using the combinatorial Gauss–Bonnet theorem and taking the means of the angles by amenability. We show that if an even-dimensional manifold is obtained from a connected sum operation from K(π, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure generalizing a result of Smillie. [ABSTRACT FROM AUTHOR]

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor g iαg¯α j are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Liouville integrable. This gives us topological obstacles to geodesic equivalence. [ABSTRACT FROM AUTHOR]

An n-dimensional submanifold X of a projective space P N( C) is called tangentially degenerate if the rank of its Gauss mapping gamma;; X → G( n, N) satisfies 0 < rank γ < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P N( C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible. [ABSTRACT FROM AUTHOR]

Using the method of moving frames, we prove that any irreducible Dupin hypersurface in S5 with four distinct principal curvatures and constant Lie curvature is equivalent by Lie sphere transformation to an isoparametric hypersurface in S5. [ABSTRACT FROM AUTHOR]