Page 350 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 350
DIFFERENTIAL GEOMETRY: OLD AND NEW (MS-15)
We described all simple (without self-intersections) closed geodesics on regular tetrahedra
in three-dimensional hyperbolic and spherical spaces. In these spaces the tetrahedron’s curva-
ture is concentrated not only into its vertices but also into its faces. The value α of the faces’
angle for hyperbolic space’s tetrahedron satisfies 0 < α < π/3 and for a tetrahedron in spheri-
cal space the faces’ angle is measured α that π/3 < α < 2π/3. The intrinsic geometry of such
tetrahedra depends on the value of its faces’ angles.
A simple closed geodesic on a tetrahedron has the type (p, q) if it has p points on each of
two opposite edges of the tetrahedron, q points on each of another two opposite edges, and there
are (p + q) points on each edges of the third pair of opposite one.
We prove 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). These geodesics exhaust all simple closed geodesics on a regular tetrahedron in
hyperbolic space. The number of simple closed geodesics of length bounded by L is asymptotic
to constant (depending on α) times L2, when L tending to infinity [1].
On a regular tetrahedron in spherical space there exists the finite number of simple closed
geodesic. The length of all these geodesics is less than 2π. For any coprime integers (p, q)
we presented the numbers α1 and α2 depending on p, q and satisfying the inequalities π/3 <
α1 < α2 < 2π/3 such that on a regular tetrahedron in spherical space with the faces’ angle
of value α ∈ (π/3, α1) there exists unique, up to the rigid motion of the tetrahedron, simple
closed geodesic of type (p, q) and on a regular tetrahedron with the faces’ angle of value α ∈
(α2, 2π/3) there is no simple closed geodesic of type (p, q).
References
[1] A A Borisenko, D D Sukhorebska, "Simple closed geodesics on regular tetrahedra in
hyperbolic space", Mat. Sb., 2020, 211(5), p.3-30. (in Russian). English translation: SB
MATH, 2020, 211(5), DOI:10.1070/SM9212
348
We described all simple (without self-intersections) closed geodesics on regular tetrahedra
in three-dimensional hyperbolic and spherical spaces. In these spaces the tetrahedron’s curva-
ture is concentrated not only into its vertices but also into its faces. The value α of the faces’
angle for hyperbolic space’s tetrahedron satisfies 0 < α < π/3 and for a tetrahedron in spheri-
cal space the faces’ angle is measured α that π/3 < α < 2π/3. The intrinsic geometry of such
tetrahedra depends on the value of its faces’ angles.
A simple closed geodesic on a tetrahedron has the type (p, q) if it has p points on each of
two opposite edges of the tetrahedron, q points on each of another two opposite edges, and there
are (p + q) points on each edges of the third pair of opposite one.
We prove 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). These geodesics exhaust all simple closed geodesics on a regular tetrahedron in
hyperbolic space. The number of simple closed geodesics of length bounded by L is asymptotic
to constant (depending on α) times L2, when L tending to infinity [1].
On a regular tetrahedron in spherical space there exists the finite number of simple closed
geodesic. The length of all these geodesics is less than 2π. For any coprime integers (p, q)
we presented the numbers α1 and α2 depending on p, q and satisfying the inequalities π/3 <
α1 < α2 < 2π/3 such that on a regular tetrahedron in spherical space with the faces’ angle
of value α ∈ (π/3, α1) there exists unique, up to the rigid motion of the tetrahedron, simple
closed geodesic of type (p, q) and on a regular tetrahedron with the faces’ angle of value α ∈
(α2, 2π/3) there is no simple closed geodesic of type (p, q).
References
[1] A A Borisenko, D D Sukhorebska, "Simple closed geodesics on regular tetrahedra in
hyperbolic space", Mat. Sb., 2020, 211(5), p.3-30. (in Russian). English translation: SB
MATH, 2020, 211(5), DOI:10.1070/SM9212
348
