Abstract
The aim of the present work is to extend Theorem 1 of the article [4] (Chap. 10) by the present author, which was there proven for polyhedra, to the more general case of two-dimensional manifolds of bounded curvature. The notion of manifold of bounded curvature was introduced by A. D. Alexandrov [1, 2], and it is defined axiomatically.
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Notes
- 1.
We say that the measures φn over R converge weakly to a measure φ0, if for any continuous function f
$$\displaystyle \begin{aligned} \int_R f(x) \mathrm{d}\varphi_0(x) = \lim_{n\to \infty} \int_R f(x) \mathrm{d}\varphi_n(x)~.\end{aligned}$$ - 2.
That is, Ln and L have parameterizations {Xn(t) : 0 ≤ t ≤ 1} and {X(t) : 0 ≤ t ≤ 1}, and \(\rho \left (X_n(t),X(t)\right )<\eta \) for all t.
References
Alexandrov, A.D.: Curves on manifolds of bounded curvature. Doklady Akad. Nauk SSSR (N.S.) 63, 349–352 (1948)
Alexandrov, A.D.: Foundations of the inner geometry of surfaces. Doklady Akad. Nauk SSSR (N.S.) 60, 1483–1486 (1948)
Alexandrov, A.D.: Selected works. Intrinsic geometry of convex surfaces. Vol. 2. Edited by S. S. Kutateladze. Transl. from the Russian by S. Vakhrameyev. Boca Raton, FL: Chapman & Hall/CRC (2005)
Reshetnyak, Y.G.: A special map** of a cone onto a polyhedron. Mat. Sb. (N.S.) 53 (95), 39–52 (1961)
Zalgaller, V.A.: On the foundations of the theory of two-dimensional manifolds of bounded curvature. Dokl. Akad. Nauk SSSR (N.S.) 108, 575–576 (1956)
Acknowledgements
First published as
[Ob odnom spetsial’nom otobrazhenii konusa v mnogoobrazie ogranichennoÄ krivizny]. SibirskiÄ MatematicheskiÄ Zhurnal, 3, 256–272 (1962). Translated from Russian by François Fillastre and Dmitriy Slutskiy.
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Reshetnyak(deceased), Y.G. (2023). On a Special Map** of a Cone in a Manifold of Bounded Curvature. In: Fillastre, F., Slutskiy, D. (eds) Reshetnyak's Theory of Subharmonic Metrics. Springer, Cham. https://doi.org/10.1007/978-3-031-24255-7_11
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DOI: https://doi.org/10.1007/978-3-031-24255-7_11
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