Sukhorebska D. Simple closed geodesics on regular tetrahedra in spaces of constant curvature

Simple closed geodesics on regular tetrahedra in three-dimensional spherical and hyperbolical spaces are studied. A combinatorial type of such geodesics, which determines the order of intersections of geodesics with the edges of tetrahedra, is introduced. Existence and uniqueness theorems for simple closed geodesics of fixed combinatorial type on regular tetrahedra, which reveal the dependence on the value of the flat angles of tetrahedra, are proved. It is shown that simple closed geodesics on a regular tetrahedron have to go through the mid-points of two pairs of edges of the tetrahedron. The asymptotic behaviour of the number of simple closed geodesics, whose length is less or equal to L, on a regular tetrahedron in the hyperbolic space is analysed as L tends to the infinity. Theorems on the extrinsic form of submanifolds with rotationally invariant metric forms in the many-dimensional hyperbolic space are proved.