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Darya Sukhorebska
PhD thesis
Title: Simple closed geodesics on regular tetrahedra in spaces of constant curvature.
Supervisor: Prof. Dr. Alexander Borisenko
Defense date: August 19, 2023
Abstract:
In Euclidean three-dimensional space the faces of a tetrahedron have zero
Gaussian curvature, and the curvature of a tetrahedron is concentrated only on its vertices.
A complete classification of closed geodesics on a regular tetrahedron
in Euclidean space follows from a tiling of Euclidean plane with regular triangles.
My dissertation studies simple closed geodesics on regular tetrahedra
in three dimensional hyperbolic and spherical spaces.
In hyperbolic or spherical space the Gaussian curvature of faces is k = -1 or 1 respectively.
Therefore, the curvature of a tetrahedron is determined not only by vertices, but also by faces.
In hyperbolic space the planar angle α of the faces of a regular tetrahedron
satisfies 0< α < π/3.
In spherical space the planar angle α satisfies π/3< α < 2π/3.
In both cases the intrinsic geometry of a tetrahedron depends on the planar angle.
The behavior of closed geodesics on a regular tetrahedron in three dimensional spaces
of constant curvature k differ depending on the sign of k.
A simple closed geodesic γ on a tetrahedron is of type (p,q)
if γ has p points on each of two opposite edges of the tetrahedron,
q points on each of two other opposite edges,
and (p+q) points on each of the remaining
two opposite edges.
We proved that on a regular tetrahedron in hyperbolic space
for any coprime integers (p,q), 0 ≤ p < q ,
there exists unique, up to the rigid motion of the tetrahedron,
simple closed geodesic of type (p,q).
This geodesic passes through the midpoints of two pairs of opposite edges of the tetrahedron.
Geodesics of type (p,q) exhaust all simple closed geodesics on a regular tetrahedron in hyperbolic space.
The number of simple closed geodesics of length bounded by L has order of grows c(α)L^2,
when L tends to infinity.
On a regular tetrahedron in spherical space
there exists the finite number of simple closed geodesics.
The length of all these geodesics is less than 2π.
For any coprime integers (p,q) we found the numbers α_1 and α_2,
satisfying the inequalities π/3< α_1 < α_2 < 2π/3, such that
1) if π/3< α α_1, then
on a regular tetrahedron in spherical space with the planar angle α
there exists the unique simple closed geodesic of type (p,q), up to the rigid motion of this tetrahedron.
This geodesic passes through the midpoints of two pairs of opposite edges of the tetrahedron;
2) if α_2 < α < 2π/3, then
on a regular tetrahedron with the planar angle α
there is no simple closed geodesic of type (p,q).
As a part of my PhD project I also considered an isometric embedding of a
submanifold of with an inner metric of revolution into hyperbolic space.
We found a condition when such a maniold admits an embedding as a submanifold of revolution.
Here you can find slides of my thesis presentation.
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Darya Sukhorebska
E-mail: suhdaria0109(at)gmail(dot)com
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