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The curvature of a space curve measures turning of its unit tangent vector; the torsion of a curve measures twisting of its osculating plane. For a regular curve $r(t)$ in Euclidean space ${\mathbb{R}}^3$,The first formula requires two derivatives and $r^{\prime }\ne 0$. The second requires three derivatives and $r^{\prime }\times r^{\prime \prime }\ne 0$: torsion of a curve is undefined by this formula at zero curvature of a space curve.

With arc length $s$ and Frenet frame $(T,N,B)$, the formulas become $\kappa =\Vert dT/ds\Vert$ and $\tau =-⟨dB/ds,N⟩$. A circle of radius $a>0$ has $\kappa =1/a$ and $\tau =0$. The circular helix $r(t)=(a\cos t,a\sin t,bt)$ has $\kappa =a/(a^2+b^2)$ and $\tau =b/(a^2+b^2)$. These data, expressed in arc length, determine the curve up to a proper Euclidean motion of Euclidean three-space by the fundamental theorem of regular space curves.^[ref]

For a compact oriented Riemannian surface $D$ with piecewise smooth boundary,Here $K$ is Gaussian curvature, $dA$ the area element of a surface, $k_g$ signed geodesic curvature, and $\chi (D)$ the Euler characteristic. Traverse each boundary component with the surface on the left. The signed exterior angle at a convex corner of interior angle ${\alpha }_j$ is ${\epsilon }_j=\pi -{\alpha }_j$. Empty boundaries contribute no boundary or corner terms.^[ref]

On a disk in the Euclidean plane, zero interior Gaussian curvature leaves total boundary turning $2\pi$. On the unit sphere, $K=1$ and $\chi =2$, giving area $4\pi$. A geodesic triangle has no smooth boundary term, giving the spherical excess formula or hyperbolic angle defect.

For a closed surface, a local metric quantity integrates to a topological invariant. A proof triangulates the surface and applies the local turning formula to each triangle. Shared-edge contributions cancel and the angle sums leave $2\pi (V-E+F)$. Thus changing the Riemannian metric cannot change total Gaussian curvature.^[ref]

Mark Powell studies the topology of manifolds, especially in dimensions three and four, together with homological algebra. His Glasgow profile verifies this research specialization.^[ref]

His Introduction to Differential Geometry by Mark Powell connects the local geometry of curves and surfaces with global invariants. Curvature and torsion of a curve describe how a curve bends in space, whereas the Gauss-Bonnet theorem turns integrated surface curvature and boundary terms into an Euler characteristic. These examples show how local differential measurements can constrain a whole geometric object.^[ref]