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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 1993 Volume 34, Number 3, Pages 199–208 (Mi smj1666)

Isometry of the boundaries of plane domains and the properties of shortest geodesics

D. A. Trotsenko


Abstract: Consider the relative metric $\rho$ in a domain $D$ of the plane. Denote by $\widetilde{D}$ the completion of $D$ with respect to the metric and denote the generalized boundary of $D$, the set $\widetilde{D}\setminus{D}$, by $\widetilde{\partial}D$. The restriction of $\rho$ onto $\widetilde{\partial}D$ is called the relative metric of the boundary.
The Main Theorem. {\it Let $D\subset\mathbb{R}^2$ be a bounded domain and $D^*\subset\mathbb{R}^2$ be an arbitrary domain. Suppose that there exists a surjective mapping $f\colon\widetilde{\partial}D\to\widetilde{\partial}D^*$ isometric in the relative metrics of the generalized boundaries $\widetilde{\partial}D$ and $\widetilde{\partial}D^*$. Then the domains $D$ and $D^*$ are isometric in Euclidean metrics and, consequently, the mapping is extandible to a Euclidean isometry of the planes.}

UDC: 517.75.1

Received: 17.04.1992


 English version:
Siberian Mathematical Journal, 1993, 34:3, 574–582

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© Steklov Math. Inst. of RAS, 2024