11 Mayıs 2017,
Mimar Sinan Güzel Sanatlar Üniversitesi Matematik Bölümü Seminerleri
On Slant Geometry
İstanbul Üniversitesi, Türkiye
It is known that a geodesic in the Poincar ́e disk is a circle which is
perpendicular to the ideal boundary (i.e., unit circle). If we adopt geodesics
as lines in the Poincar ́e disk, we have the model of Hyperbolic geometry. We
have another class of curves in the Poincar ́e disk which has an analogous
property with lines in the Euclidean plane. A horocycle is a circle which is
tangent to the ideal boundary. We note that a line in the Euclidean plane
can be considered as the limit of the circles when the radius tends to infinity.
In the same manner, a horocycle is also a curve which can be considered
as the limit of the circles in the Poincar ́e disk when the radius tends to
infinity. Hence, horocycles are also an analogous notion of lines. If we adopt
horocycles as lines, what kind of geometry do we obtain? We say that
two horocycles are parallel if they have common tangent point at the ideal
boundary. Under this definition, the parallel axiom is satisfied. However, for
any two fixed points in the Poincar ́e disk, there exist always two horocycles
passing through these points, so that the first axiom of Euclidean geometry
is not satisfied. In the case of general dimensions, this geometry is said
to be horospherical geometry. On the other hand, we have another kind of
curves in the Poincar ́e disk which has similar properties with Euclidean lines.
An equidistant curve is a circle whose intersection with the ideal boundary
consists of two points. Generally, the angle between an equidistant curve and
the ideal boundary is φ ∈ (0, π/2]. Here, we emphasize that a geodesic is an
equidistant curve with φ = π/2. However, a horocycle is not an equidistant
curve, but it is a circle with φ = 0. We call the geometry where φ = π/2
vertical geometry and the geometry where φ = 0 horizontal geometry. And
also we call the family of geometry depending on φ slant geometry.
In this talk, I will give some of recent results about this geometry including joint works with Mikuri Asayama, Shyuichi Izumiya and Aiko Tamaoki.
Seminar room, Bomonti Campus, MSGSÜ