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Register a new userThe adjacent sides of hyperbolic Lambert quadrilaterals
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Authors
Gendi Wang
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We prove sharp bounds for the product and the sum of the hyperbolic lengths of a pair of hyperbolic adjacent sides of hyperbolic Lambert quadrilaterals in the unit disk. We also show the Holder convexity of the inverse hyperbolic sine function involved in the hyperbolic geometry.
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We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal mappings from the unit disk onto itself and obtain sharp results in this case, too.
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are euclidean or hyperbolic lines. In the case of hyperbolic lines, this type of quadrilaterals are called ideal quadrilaterals. Our main result gives a euclidean counterpart of an earlier result on the hyperbolic distances between the opposite sides of ideal quadrilaterals. The proof is based on computations involving hyperbolic geometry. We also found a new formula for the hyperbolic midpoint of a hyperbolic geodesic segment in the unit disk. As an application of some geometric properties, we provided a euclidean construction of the symmetrization of random four points on the unit circle with respect to a diameter which preserves the absolute cross ratio of quadruples.
We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.
A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into the hyperbolic plane is achieved.
Given a domain $G subsetneq Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D subset G$. In each case, distances in the subdomain are always larger than in the original domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.
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