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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Simplexes in Riemannian manifolds

Author: B. V. Dekster
Journal: Proc. Amer. Math. Soc. 118 (1993), 1227-1236
MSC: Primary 52A55; Secondary 53C99
MathSciNet review: 1136234
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Abstract: Existence of a simplex with prescribed edge lengths in Euclidean, spherical, and hyperbolic spaces was studied recently. A simple sufficient condition of this existence is, roughly speaking, that the lengths do not differ too much. We extend these results to Riemannian $ n$-manifolds $ {M^n}$. More precisely we consider $ m + 1$ points $ {p_0},{p_1}, \ldots ,{p_m}$ in $ {M^n},m \leqslant n$, with prescribed mutual distances $ {l_{ij}}$ and establish a condition on the matrix $ ({l_{ij}})$ under which the points $ {p_i}$ can be selected as freely as in $ {R^n}:{p_0}$ is a prescribed point, the shortest path $ {p_0}{p_1}$ has a prescribed direction at $ {p_0}$, the triangle $ {p_0}{p_1}{p_2}$ determines a prescribed $ 2$-dimensional direction at $ {p_0}$, and so on.

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Keywords: Simplexes, mutual distances between points, comparison theorems for triangles
Article copyright: © Copyright 1993 American Mathematical Society

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