Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities). (French) Zbl 0611.53035

Troyanov, Marc

Enseign. Math., II. Sér. 32, 79-94 (1986).

A Euclidean surface with cone singularities is a surface which locally has the structure of a standard cone, and a structure cone is determined by the cone angle \(\theta\). Of particular interest are the examples of such structures which arise from quadratic differentials on a Riemannian surface. On a surface S with n singularities, it follows from the Gauss- Bonnet formula that \[ \sum^{n}_{i=1}(2\pi -\theta_ i)=2\pi \chi (S). \] The author gives a clear and elementary account of such surfaces and proves the classification theorem that gives a closed orientable surface S with n distinguished points and prescribing angles \(\theta_ 1,...,\theta_ n\) satisfying the above equation then in each class of conformal structures on S there is a suitably-normalized unique Euclidean structure with singularities with given cone angles at the distinguished points.

Reviewer: C.Maclachlan

MSC:

53C20	Global Riemannian geometry, including pinching
53A30	Conformal differential geometry (MSC2010)

Keywords:

cone singularity; flat surface; structure cone; cone angle; quadratic differentials; Riemannian surface; conformal structures