ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

Trigonometrical Identities and Isoperimetric Inequality

Ruslan Shmatkov Sobolev Institute of Matehematics, Novosibirsk, Russia

The main result of this report is the following:

Theorem 1. Let 2 < m_{0}, n_{0} < 4 real roots of the equation f(m_{0}, n_{0}) = 0, where

f(m, n) = −2 + 6(M_{m} + M_{n}) − 6(M_{m}^{2} + M_{n}^{2}) + 2(M_{m}^{3} + M_{n}^{3}) + 65M_{m}M_{n} +
32(M_{m}^{2}M_{n} + M_{m}M_{n}^{2}) + 26M_{m}^{2}M_{n}^{2} + 5(M_{m}^{3}M_{n} + M_{m}M_{n}^{3}) +
4(M_{m}^{3}M_{n}^{2} + M_{m}^{2}M_{n}^{3}) + M_{m}^{3}M_{n}^{3}

and M_{k} = cot^{2}(π/k). Then there exists a 32-faced polyhedron in the 3-dimentional Euclidean space which is the canonical fundamental set of the Euclidean cone-manifold W (m_{0}, n_{0}).

A.D. Mednykh (1999) established the Tangent and the Sine Rules relating the complex lengthes of the singular geodesics and the cone angles of W (m, n) (Theorems A and B).

In this report the Euclidean analogues of the Theorems A and B are obtained.

Theorem 2 (The Euclidean Tangent Rule). Let γ_{m} = l_{m} + i φ_{m} be a complex length of the singular geodesic of the hyperbolic cone-manifold W (m, n) with cone angle 2π/m. If W (m, n) admits the Euclidean structure, then

\begin{matrix}tan & \frac{φ_{m}}{4} \\ tan & \frac{φ_{n}}{4} \end{matrix} = \begin{matrix}tan & \frac{π}{m} \\ tan & \frac{π}{n} \end{matrix} .

Theorem 3 (The Euclidean Sine Rule I). Let γ_{m} = l_{m} + i φ_{m} be a complex length of the singular geodesic of the hyperbolic cone-manifold W (m, n) with cone angle 2π/m. If W (m, n) admits the Euclidean structure, then

\frac{sin \frac{φ_{m}}{2}}{l_{m}} = \frac{sin \frac{φ_{n}}{2}}{l_{n}} .

Moreover, in this report the following theorems are established.

Theorem 4 (The Euclidean Sine Rule II). Under condidions of Theorem 3 we have

\begin{matrix}sin & (\frac{2π}{m} − \frac{φ_{m}}{2} ) \\ sin & \frac{φ_{n}}{2} \end{matrix} = \begin{matrix}sin & (\frac{2π}{n} − \frac{φ_{n}}{2} ) \\ sin & \frac{φ_{m}}{2} \end{matrix}.

Theorem 5. If W (m, n) admits the Euclidean structure, then

V ol W (m, n) = \frac{1}{12} (\frac{4 − sin^{2} θ}{sin θ}) l_{m}l_{n}a,

where V ol W (m, n) is the volume, a is the shortest distance and θ is the acute angle between the components of the singular set of W (m, n).

Theorem 6 (Isoperimetric Inequality). Under conditions of Theorem 5 we have

0 < V ol W (m, n) < 0, 29 l_{m}l_{n}a.

This report is supported by RFBR (grant 99-01-00630).