Differential Geometry of Curves and Surfaces

Differential geometry studies smooth shapes using calculus. When the objects are onedimensional curves ortwodimensional surfaces embedded in Euclidean space, the central ideas are curvature, torsion, and thefundamental forms. These concepts allow us to quantify how a curve bends, how a surface bends in differentdirections, and how local geometry relates to global properties.

1. Curves in \(\mathbb{R}^3\)

A regular curve is a smooth map \(\gamma : I\subset\mathbb{R}\rightarrow\mathbb{R}^3\) with nonvanishing velocity\(\gamma'(t)\). By reparametrising by arc length,\[s(t)=\int_{t_0}^{t}\!\|\gamma'(\tau)\|\,d\tau,\]we obtain a unitspeed curve \(\mathbf{r}(s)\) satisfying \(\|\mathbf{r}'(s)\|=1\). The unit tangent vector is\(\mathbf{T}(s)=\mathbf{r}'(s)\). Differentiating once more gives the curvature vector\[\mathbf{r}''(s)=\kappa(s)\,\mathbf{N}(s),\]where \(\kappa= \|\mathbf{r}''(s)\|\) is the curvature and \(\mathbf{N}\) the principal normal (a unit vectororthogonal to \(\mathbf{T}\)). The binormal \(\mathbf{B}=\mathbf{T}\times\mathbf{N}\) completes an orthonormal frameknown as the FrenetSerret frame.

FrenetSerret Formulas

\[\begin{aligned}\mathbf{T}'(s) &= \kappa(s)\,\mathbf{N}(s),\\[4pt]\mathbf{N}'(s) &= -\kappa(s)\,\mathbf{T}(s)+\tau(s)\,\mathbf{B}(s),\\[4pt]\mathbf{B}'(s) &= -\tau(s)\,\mathbf{N}(s),\end{aligned}\]where \(\tau\) is the torsion, measuring how the osculating plane twists along the curve.

The curvature \(\kappa\) is intrinsic to the curves shape: circles have constant curvature \(1/R\),while straight lines have \(\kappa=0\). Torsion distinguishes a helix from a planar curve that share the samecurvature profile.

2. Surfaces in \(\mathbb{R}^3\)

A regular surface is given locally by a parametrisation\[\mathbf{X}(u,v):U\subset\mathbb{R}^2\rightarrow\mathbb{R}^3,\]with linearly independent tangent vectors \(\mathbf{X}_u,\mathbf{X}_v\). The tangent plane at a point\(p=\mathbf{X}(u_0,v_0)\) is spanned by these vectors. The normal vector is defined (up to sign) by\[\mathbf{N}=\frac{\mathbf{X}_u\times\mathbf{X}_v}{\|\mathbf{X}_u\times\mathbf{X}_v\|}.\]The differential geometry of a surface is encoded in its first and second fundamental forms.

First Fundamental Form

\[I = E\,du^2+2F\,du\,dv+G\,dv^2,\quad\text{where}\quad E=\langle\mathbf{X}_u,\mathbf{X}_u\rangle,\;F=\langle\mathbf{X}_u,\mathbf{X}_v\rangle,\;G=\langle\mathbf{X}_v,\mathbf{X}_v\rangle.\]

\(I\) measures infinitesimal lengths on the surface; the induced metric determines angles, areas,and geodesic distances. The area element is \(\mathrm{d}A=\sqrt{EG-F^2}\,du\,dv\).

Second Fundamental Form

\[II = L\,du^2+2M\,du\,dv+N\,dv^2,\quad\text{with}\quad L=\langle\mathbf{X}_{uu},\mathbf{N}\rangle,\;M=\langle\mathbf{X}_{uv},\mathbf{N}\rangle,\;N=\langle\mathbf{X}_{vv},\mathbf{N}\rangle.\]

\(II\) captures how the surface curves in space. The shape operator (Weingarten map) \(S\) satisfies\(II(\mathbf{w},\mathbf{v}) = \langle S\mathbf{w},\mathbf{v}\rangle\) for tangent vectors \(\mathbf{w},\mathbf{v}\). Its eigenvalues\(k_1,k_2\) are the principal curvatures.

Gaussian and Mean Curvature

\[K = k_1k_2 = \frac{LN-M^2}{EG-F^2},\qquadH = \frac{k_1+k_2}{2} = \frac{EN-2FM+GL}{2(EG-F^2)}.\]

The Gaussian curvature \(K\) is intrinsic: it depends only on the metric (first fundamental form) and isunchanged by isometries. The mean curvature \(H\) reflects extrinsic bending; the surface of a soap filmsatisfies \(H=0\) (minimal surface).

3. Theorema Egregium

Gausss remarkable theorem states that the Gaussian curvature can be computed solely from the firstfundamental form and its derivatives. Consequently, a surface can be flattened without distortion onlywhen \(K=0\) everywhere (developable surfaces such as cylinders and cones). A sphere, with constant\(K>0\), cannot be mapped isometrically onto a plane.

4. Examples

Surface	Parametrisation	Curvatures
Plane	\(\mathbf{X}(u,v)=(u,v,0)\)	\(K=0,\;H=0\)
Cylinder	\(\mathbf{X}(u,v)=(\cos u,\sin u, v)\)	\(K=0,\;H=\frac{1}{2R}\)
Sphere (radius \(R\))	\(\mathbf{X}(u,v)=(R\sin v\cos u, R\sin v\sin u, R\cos v)\)	\(K=\frac{1}{R^2},\;H=\frac{1}{R}\)
Helicoid	\(\mathbf{X}(u,v)=(v\cos u,\;v\sin u,\;cu)\)	\(K=-\frac{c^2}{(c^2+v^2)^2},\;H=0\)

The helicoids zero mean curvature demonstrates that it is a minimal surface, while its negative Gaussiancurvature shows a saddle shape at every point.

5. Geodesics

A geodesic is a curve on a surface that locally minimises length; equivalently, its acceleration is normalto the surface. In local coordinates the geodesic equations are\[\frac{d^2x^i}{ds^2}+\Gamma^i_{jk}\frac{dx^j}{ds}\frac{dx^k}{ds}=0,\]where \(\Gamma^i_{jk}\) are the Christoffel symbols derived from the first fundamental form. Great circles on a sphereand straight lines on a plane are familiar examples.

6. Applications

• Computer graphics: curvature information guides shading, mesh refinement, and surfacefitting.
• General relativity: the curvature of spacetime, expressed by the Riemann tensor, generalises Gaussian curvature.
• Robotics and path planning: geodesic concepts are used for optimal motion on curved terrains.
• Architecture: minimal surfaces inspire lightweight, structurally efficient roofs.

7. Further Reading

For a deeper treatment, consider classic texts such as Elementary Differential Geometry byBarrett O'Neill, Differential Geometry of Curves and Surfaces by Manfredo DoCarmo, andthe modern exposition in Riemannian Geometry by Petersen. Online resources, includingthe Wikipedia article,provide additional examples and visualisations.