Regular oriented surface regions with corners

Let $\mathrm{\Sigma }$ be an oriented smooth surface without boundary and $D\subseteq \mathrm{\Sigma }$ a compact subset; $\mathrm{\partial }D$ means its topological boundary in $\mathrm{\Sigma }$. Call $D$ a regular oriented surface region if $D=\overline{{\mathrm{Int}}_{\mathrm{\Sigma }}D}$ and its boundary, with the following supplied finite decomposition, is a finite disjoint union of simple closed curves. Each curve is a cyclic concatenation of finitely many regular $C^2$ embedded arcs; the arc images meet only at their designated consecutive endpoints. Away from those endpoints, a boundary chart identifies $D$ locally with a closed half-disk bounded by the corresponding $C^2$ arc. At an endpoint (a vertex), a local chart identifies $D$ with the closure of one sector bounded by the two incident arcs. If $v_{-}$ and $v_{+}$ are the nonzero one-sided velocities directed into and out of the vertex along the cyclic parametrization, require $v_{+}\mathrm{\ne }-c{v}_{-}$ for every $c>0$. Thus the two rays bounding the sector are distinct: the cusp whose exterior turn would be $\pm \pi$ is excluded. Smooth subdivision points may be removed from the decomposition; the empty boundary is allowed. The chart conditions are part of the definition: a piecewise-smooth closed curve is not declared to bound a region unless a domain side with these local models is supplied.

On each smooth boundary arc orient its tangent line by the outward-normal-first rule: for an outward transverse vector $\nu$, the selected tangent direction $T$ is the one for which $(\nu ,T)$ is positive in $T\mathrm{\Sigma }$. At a vertex this orientation is understood by its one-sided limits on the two arcs; no tangent at the vertex is part of the data. A disk region is a nonempty connected regular region supplied with a homeomorphism of pairs $(\overline{B^2},S^1)\to (D,\mathrm{\partial }D)$, where $\overline{B^2}$ is the closed unit disk.

The wedge chart allows either a convex or a reflex sector. Its boundary arcs are $C^2$ separately; the definition does not assert $C^2$ smoothness across a vertex. The disk homeomorphism is topological data and imposes no additional smoothness at the vertices.

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, §“Plane Geometry,” printed pp. 157–159, defines simple closed piecewise-smooth curves, finite vertices, positive direction, and excludes cusps by excluding exterior angles $\pm \pi$; §“The Gauss–Bonnet Formula,” printed pp. 162–163, carries these conventions to oriented surface regions. Datar, Lectures on Riemannian Geometry, Lecture 1, Definitions 1.1.1 and 1.2.1, printed pp. 3–5, and Lecture 2, §2.0, printed pp. 10–11, gives the corresponding regular-curve, curved polygon, no-cusp, and positive-orientation conventions. The explicit chart and disk-topology clauses above make the domain hypotheses precise; the boundary orientation calculation is checked above from the cited library definition.