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Using the two first fundamental forms calculated below, chooseThe surface of revolution is then a catenoid, with metric ${\cosh }^{2}s(d{s}^2+d{v}^2)$. On the helicoid, take $u=\sinh s/a$, assuming $a\ne 0$ so that its parametrization is regular. Substitution givesThus the coordinate mapis a local isometry on sufficiently small patches. The constant $a$ only changes the radial parametrization of the given helicoid, whose pitch is fixed at one. For $a=0$ the displayed helicoid parametrization is not a regular surface.

Two regular coordinate patches are locally isometric under a coordinate correspondence exactly when it preserves their first fundamental forms. More explicitly, a diffeomorphism $\Phi :U\to \tilde{U}$ of parameter domains induces a local isometry if and only ifIn common coordinates this means equality of the three coefficient functions $E,F,G$. The criterion follows because the first fundamental form gives the inner product of every pair of tangent vectors. Equality therefore preserves lengths of all curves; conversely a smooth isometry preserves tangent-vector lengths and their inner products by polarization.

For a regular parametrization $\mathbf{r}(u,v)$, the first fundamental form is the induced Riemannian metricIt measures squared lengths of tangent vectors and therefore lengths of curves. In the plane put $x=\rho \cos \theta$, $y=\rho \sin \theta$. The derivatives are orthogonal, of lengths one and $\rho$, soThese polar coordinates are regular for $\rho >0$.

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, let $n$ be its degree and $e_p$ its local degree at $p$. The Riemann-Hurwitz formula isEquivalently, $2g_X-2=n(2g_Y-2)+{\sum }_p(e_p-1)$. Only finitely many terms are nonzero; their sum is the total ramification. In local coordinates the map has the form $w=z^{e_p}$ after suitable coordinate changes.

Form the dual graph of the finite triangulation: its vertices are the triangles, and two vertices are joined when their triangles share an edge. Connectedness of the surface makes this graph connected. Choose a spanning tree and begin with one disjoint triangle. Attach the remaining triangles along the tree edges, one at a time, always joining a new triangle along one boundary edge. At every step the result is a topological disk, so the final union is a polygonal disk.

The edges not used for these attachments still occur twice on the boundary, once for each incident triangle. Identify each pair as in the original triangulation. The result recovers the original surface. This proves a compact connected closed surface is a polygon with its boundary edges identified in pairs. Closedness is the convention here; a boundary would leave some edges unpaired.

For a finite CW complex with $V$ vertices, $E$ edges and $F$ faces,This Euler characteristic is unchanged by subdividing an edge, which adds one vertex and one edge, or subdividing a face by an edge, which adds one edge and one face. Common refinements therefore give the same value for different subdivisions of the same surface.