## Intrinsic geometry of surfaces |

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### Contents

The Simplest Concepts and the Object of the Investigation | 1 |

Twodimensional manifolds of bounded curvature | 4 |

Approximation by polyhedral metrics | 7 |

Copyright | |

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### Common terms and phrases

absolute curvature absolutely convex adjacent analogous arbitrary assertion Borel sets boundary bounded curvature bounded variation broken curves Chapter coincide compact complete angle consider construction convex polygon corresponding decomposed decomposition decrease defined disc distance endpoints equal equation estimate excesses excised exists Figure finite number finite system follows Gauss-Bonnet theorem geodesic inequality interior point intersections intrinsic geometry intrinsic metric leftmost shortest arc Lemma Lemma 15 length lim inf lim sup lying manifold of bounded metric space Moreover negative curvature obtain open set pass shortest arcs pasting pieces plane triangle polygonal curves polygons Q polyhedral metric positive curvature proof of Theorem proved rectifiable curve reduced triangles region G right rotation rotation of bounded satisfied sector angles segment sequence set G shortest arc XY simple arc simple polygon simple triangles subdivision subsection Suppose Theorem 9 triangles homeomorphic two-dimensional manifold two-gon uniform convergence uniformly bounded upper angle vertex weak convergence zero