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The set Homeo(M) of all homeomorphisms of a manifold M onto itself is the group related to composition and inverse mapping. The group of homeomorphisms of smooth manifolds is of great importance in differential geometry and in analysis. It is known that the group Homeo(M) is a topological group in compact open topology. In this paper we investigate the group HomeoF (M) of homeomorphisms foliated manifold (M, F) with foliated compact open topology. It is proven that foliated compact open topology of the group HomeoF (M) has a countable base. It is also proven that the group HomeoF (M) is a topological group with foliated compact open topology. [ABSTRACT FROM AUTHOR]

In this paper, we study convex polyhedra that are uniquely projected onto the XOY plane and one of the edges is perpendicular to vector e → (O X) , but do not have support planes perpendicular to vector e → (O X) , and it is shown that the conditional total angle of such polyhedral angles is bounded. This conditional full angle is shown to be invariant with respect to a given linear transformation with a non-orthogonal and non-symmetric matrix, but whose determinant is equal to one. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of the development of convex polyhedra that preserves the isometry on sections, for such polyhedra an invariant associated with the vertex of a convex polyhedral angle. For such polyhedra an invariant associated with the vertex of a convex polyhedral angle is found. In one of the directions in classical differential geometry, the properties of geometric objects are studied along their entire range, which is called "geometry in large". [ABSTRACT FROM AUTHOR]