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The first fundamental form of an embedded surface $S\subseteq {\mathbb{R}}^3$ is the inner product induced on each tangent plane:For a tangent vector field $V$ along a curve $\alpha$, its surface covariant derivative isthe tangential projection of its ordinary derivative. A curve is a geodesic when

A local isometry preserves the first fundamental form, so its Christoffel symbols and Levi--Civita covariant derivatives satisfyTaking $V={\alpha }^{\prime }$ shows that if $\alpha$ is geodesic, then $\varphi \circ \alpha$ is geodesic.

The converse is false. The dilationmaps every affinely parametrized straight-line geodesic to another such geodesic, butso it does not preserve the first fundamental form. This is the geodesic-preserving homothety that is not a local isometry.

For a positively oriented local parametrization $X(u,v)$ with first fundamental formthe area element of a surface isUnder an orientation-preserving coordinate change, the Jacobian from $du\wedge dv$ cancels the inverse Jacobian in the square root of the metric determinant. Hence these local expressions agree and define a global two-form.

The Euler characteristic may be defined from any finite triangulation byor equivalently by the alternating sum of the dimensions of the rational homology groups. Subdivision leaves $V-E+F$ unchanged, and the homological formula shows that it is a topological invariant, so the definition does not depend on the triangulation.

Give the boundary its induced orientation and parametrize it by arc length. If $T$ is its unit tangent and $N$ is the chosen unit normal to the surface, its signed geodesic curvature iswhere $D_s$ is the surface covariant derivative. The Gauss-Bonnet theorem for a compact oriented surface with smooth boundary and no corners iswhere $K$ is the Gaussian curvature.

An orientation of $S$ chooses a unit normal at every point. The Gauss map isFor the unit tangent $T=\dot{\alpha }$, the surface covariant derivative is the tangential projectionThe signed geodesic curvature isBecause $|T|=1$, $D_{s}T$ is perpendicular to $T$ and henceConsequently ${\kappa }_g=0$ means $D_{s}T=0$: the tangent is parallel along $\alpha$, so $\alpha$ is a geodesic.