Triangulations presque équilatérales des surfaces. (Almost equilateral triangulations of surfaces) (Q917942)

The paper concerns the existence of some special geodesic triangulations of compact Riemannian surfaces without boundary. Let \(0<\alpha \leq \beta\) and let \((T_ n)\) be a sequence of geodesic triangulations of a compact Riemannian surface (X,g). The sequence \((T_ n)\) is said to be an \(\{\alpha\),\(\beta\}\)-triangulation of (X,g) whenever (i) sup\(\{\) diam T; \(T\in T_ n\}\to 0\) when \(n\to \infty,\) (ii) for every \(\epsilon >0\) there exists \(n_ 0\) such that for \(n\geq n_ 0\) the angles of all the triangles of \(T_ n\) belong to \([\alpha - \epsilon,\beta +\epsilon].\) We say that (X,g) is \(\{\alpha\),\(\beta\}\)-triangulable if there exists such a sequence \((T_ n)\). The main result is the following: If \(X=S^ 2\), then (X,g) is \(\{\) \(3\pi\) /10, \(2\pi\) /5\(\}\)-triangulable for every metric g. If X is a torus, then (X,g) is \(\{\pi\) /3, \(\pi\) /3\(\}\)- triangulable for every g. If X is an orientable compact surface of genus at least 2, then (X,g) is \(\{\) \(2\pi\) /7,5\(\pi\) /14\(\}\)-triangulable for every g. All these estimates are optimal. [In the reviewer's opinion, the redaction of this paper is rather strange.]