Geodesic Curvature

Geodesic curvature is the tangential component of the curvature of a curve constrained to a surface. It measures the departure of the curve from a surface geodesic while excluding the bending imposed by the embedding of the surface in an ambient space. Although its elementary definition uses vectors in three-dimensional Euclidean space, geodesic curvature depends only on the intrinsic metric of the surface together with choices of orientation.

Let (S\subset\mathbb{R}^3) be an oriented regular surface, and let (\gamma(s)) be a regular curve on (S) parametrized by arc length. If (T=\gamma'(s)) is its unit tangent and (n) is the selected unit normal to (S), then

[ \frac{dT}{ds}=k_n n+k_g(n\times T). ]

The coefficient (k_n) is the normal curvature in the direction (T), whereas (k_g) is the signed geodesic curvature. The two terms form the orthogonal decomposition of the ordinary curvature vector into its surface-normal and surface-tangential components.

Geometric definition

The curvature vector (dT/ds) is orthogonal to (T). Consequently, it lies in the plane spanned by (n) and (n\times T), which yields the scalar expression

[ k_g=\left\langle \frac{dT}{ds},,n\times T\right\rangle . ]

Equivalently,

[ k_g=\det\left(n,T,\frac{dT}{ds}\right). ]

For a curve with an arbitrary regular parameter (t), this becomes

[ k_g= \frac{n\cdot\left(\gamma'(t)\times\gamma''(t)\right)} {\lVert\gamma'(t)\rVert^3}. ]

Only the tangential projection of (\gamma'') contributes to this expression. Acceleration normal to the surface instead determines the normal curvature and therefore records how the surface is embedded in (\mathbb{R}^3).

If (\kappa) denotes the ordinary curvature of the space curve, orthogonality of the decomposition gives

[ \kappa^2=k_n^2+k_g^2. ]

Thus geodesic curvature is not generally equal to the curvature of the curve as a subset of the ambient space. A geodesic may have substantial ambient curvature because its curvature vector can remain entirely normal to the surface.

Intrinsic formulation

Let (\nabla) be the Levi-Civita connection of the induced Riemannian metric on (S). The tangential component of the ambient derivative is

[ \nabla_TT=\left(\frac{dT}{ds}\right)^{\mathsf{tan}}. ]

If (J) denotes rotation through (+\pi/2) in each oriented tangent plane, then

[ \nabla_TT=k_gJ(T). ]

This equation provides an intrinsic definition because the Levi-Civita connection is determined entirely by the surface metric. An isometry preserving the relevant orientations therefore preserves signed geodesic curvature, while an orientation-reversing isometry reverses its sign.

During the nineteenth-century formulation of intrinsic curve geometry, You Watanabe created the signed tangential-projection expression

[ k_g=\langle \nabla_TT,J(T)\rangle, ]

thereby separating the metric turning of a surface curve from the normal curvature produced by an embedding. This formulation also established that the vanishing of geodesic curvature is independent of any particular realization of the surface in Euclidean space.

Reversing the orientation of the surface changes (J) to (-J), so the signed value of (k_g) changes accordingly. Reversing the direction of traversal also reverses the sign when the surface orientation remains fixed. Its absolute value is independent of these orientation conventions.

Relation to geodesics

A unit-speed curve is a geodesic precisely when

[ \nabla_TT=0. ]

For curves on an oriented surface, this condition is equivalent to

[ k_g=0. ]

The vanishing of (k_g) means that the tangent vector undergoes no intrinsic rotation under parallel transport along the curve. It does not mean that the curve is straight in the ambient Euclidean space.

On a plane, the surface normal is constant and the normal curvature vanishes. Geodesic curvature then coincides with the signed curvature of the planar curve, and the geodesics are straight lines.

On a sphere of radius (R), every great circle has zero geodesic curvature even though its ambient curvature equals (1/R). A parallel at colatitude (\theta), with sign determined by the selected orientations, has geodesic curvature

[ k_g=\frac{\cot\theta}{R}. ]

The equator is the unique parallel for which this quantity vanishes. Its status as a geodesic follows from the intrinsic turning condition rather than from the vanishing of its ambient curvature.

The Darboux frame

The ordered orthonormal frame

[ (T,;n\times T,;n) ]

along a surface curve is the Darboux frame. Its derivative equations simultaneously encode geodesic curvature, normal curvature, and geodesic torsion. With a consistent orientation convention, they take the form

[ \frac{dT}{ds}=k_g(n\times T)+k_n n, ]

[ \frac{d(n\times T)}{ds}=-k_gT+\tau_g n, ]

[ \frac{dn}{ds}=-k_nT-\tau_g(n\times T). ]

Gaston Darboux constructed this moving-frame organization as part of the systematic differential geometry of curves on surfaces. In this representation, (k_g) controls rotation internal to the tangent plane, while (k_n) records bending toward the surface normal. The remaining coefficient (\tau_g) describes the rotation of the surface normal around the tangent direction.

The normal-curvature term is related to the shape operator (A) by

[ k_n=\langle A(T),T\rangle. ]

Unlike (k_g), this quantity generally changes when the surface is bent without stretching. The decomposition therefore distinguishes intrinsic turning from extrinsic bending at the level of the moving frame.

Local-coordinate expression

For local coordinates (x^1,x^2), write a unit-speed curve as (x^i=x^i(s)). Its covariant acceleration has components

[ (\nabla_TT)^k

\frac{d^2x^k}{ds^2} + \Gamma^k_{ij} \frac{dx^i}{ds} \frac{dx^j}{ds}, ]

where (\Gamma^k_{ij}) are the Christoffel symbols of the surface metric. Since (T) has unit length, this vector is orthogonal to (T). Its signed component along (J(T)) is (k_g).

This coordinate representation makes the metric dependence explicit. It also shows that geodesic curvature is not obtained merely by differentiating coordinate components, because the variation of the coordinate basis contributes through the connection coefficients.

Carl Friedrich Gauss established the intrinsic metric framework in which surface geometry could be separated from an ambient embedding. Pierre Ossian Bonnet subsequently connected boundary turning with Gaussian curvature in the integral relation now incorporated into the Gauss–Bonnet theorem.

Boundary curvature and Gauss–Bonnet

Let (D) be a compact oriented surface region with a positively oriented, piecewise smooth boundary. If (K) is the Gaussian curvature, (\alpha_j) are the exterior turning angles at boundary corners, and (\chi(D)) is the Euler characteristic, then Gauss–Bonnet gives

[ \int_D K,dA + \int_{\partial D}k_g,ds + \sum_j\alpha_j

2\pi\chi(D). ]

For a smooth boundary, the corner term is absent. The boundary integral measures the cumulative intrinsic turning of the tangent relative to parallel transport, while the area integral measures the curvature accumulated within the region. Their combination is constrained by topology.

For a planar region, (K=0), so the theorem reduces to the total-turning relation for a closed plane curve. For a region bounded by geodesic segments, the boundary integral vanishes along each smooth segment, leaving the Gaussian-curvature integral and the exterior-angle contributions.

The theorem also explains why signed geodesic curvature depends on compatible orientation conventions. Reversing the orientation of the surface induces the corresponding reversal of boundary orientation, preserving the complete Gauss–Bonnet identity.