I am currently working on a topology project which involves proving the following theorem.
Let $M$ be a 2-manifold defined as the quotient space of $\mathbb{R}^2$ by some gluing ~ imposed on a fundamental domain $O\subset\mathbb{R}^2$. If
$\cdot$ $M$ has no cone points
$\cdot$ $O$ is a Euclidean polygon
$\cdot$ ~ is a Euclidean gluing
Then $M$ has flat geometry.
I have had more than one conversation with my professor about what is meant by these conditions and am yet to find a satisfying, formal definition for literally anything here. Can anyone provide me with a definition or (ideally) a source which gives more formal definitions of cone points, euclidean polygons, and euclidean gluings? He has described all of these things to me in very loose terminology; e.g. cone points are where loops on the manifold do not turn through 2$\pi$ on the fundamental domain, and euclidean gluings are ones which act like isometries on the boundaries.
He has given me a more formal definition of flat geometry in the sense that $M$ shares the structure of $(\mathbb{R}^2,\operatorname{Isom}(\mathbb{R}^2))$ in that there exists an atlas covering $M$ so that the transition map between every two charts in $M$ is an isometry.
If anyone could give me some help on where to look to put some more formality to these notions, even if it's just a nudge in one direction, it would be much appreciated.
If you read French, start by reading
M. Troyanov, Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2) 32 (1986), no. 1-2, 79–94.
(That paper deals with the locally Euclidean case you are interested in.) Otherwise, start by reading
M. Troyanov, Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821,
which is more general.