In Riemannian geometry, the geodesic curvature {displaystyle k_{g}} of a curve {displaystyle gamma } measures how far the curve is from being a geodesic. In a given manifold {displaystyle {bar {M}}}, the geodesic curvature is just the usual curvature of {displaystyle gamma } (see below), but when {displaystyle gamma } is restricted to lie on a submanifold {displaystyle M} of {displaystyle {bar {M}}} (e.g. for curves on surfaces), geodesic curvature refers to the curvature of {displaystyle gamma }in {displaystyle M} and it is different in general from the curvature of {displaystyle gamma } in the ambient manifold {displaystyle {bar {M}}}. The (ambient) curvature {displaystyle k} of {displaystyle gamma } depends on two factors: the curvature of the submanifold {displaystyle M} in the direction of {displaystyle gamma } (the normal curvature {displaystyle k_{n}}), which depends only on the direction of the curve, and the curvature of {displaystyle gamma } seen in {displaystyle M} (the geodesic curvature {displaystyle k_{g}}), which is a second order quantity. The relation between these is {displaystyle k={sqrt {k_{g}^{2}+k_{n}^{2}}}}. In particular geodesics on {displaystyle M} have zero geodesic curvature (they are "straight"), so that {displaystyle k=k_{n}}, which explains why they appear to be curved in ambient space whenever the submanifold is.

Definition

Consider a curve {displaystyle gamma } in a manifold {displaystyle {bar {M}}}, parametrized by arclength, with unit tangent vector {displaystyle T=dgamma /ds}. Its curvature is the norm of the covariant derivative of {displaystyle T}: {displaystyle k=|DT/ds|}. If{displaystyle gamma } lies on {displaystyle M}, the geodesic curvature is the norm of the projection of the covariant derivative {displaystyle DT/ds} on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of {displaystyle DT/ds} on the normal bundle to the submanifold at the point considered.

If the ambient manifold is the euclidean space {displaystyle mathbb {R} ^{n}}, then the covariant derivative {displaystyle DT/ds}is just the usual derivative {displaystyle dT/ds}.

Let {displaystyle M} be the unit sphere {displaystyle S^{2}} in three-dimensional Euclidean space. The normal curvature of {displaystyle S^{2}} is identically 1, independently of the direction considered. Great circles have curvature {displaystyle k=1}, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius {displaystyle r} will have curvature {displaystyle 1/r} and geodesic curvature {displaystyle k_{g}={frac {sqrt {1-r^{2}}}{r}}}.

• The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold {displaystyle M}. It does not depend on the way the submanifold {displaystyle M} sits in {displaystyle {bar {M}}}.

• Geodesics of {displaystyle M} have zero geodesic curvature, which is equivalent to saying that {displaystyle DT/ds} is orthogonal to the tangent space to {displaystyle M}.

• On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: {displaystyle k_{n}} only depends on the point on the submanifold and the direction {displaystyle T}, but not on {displaystyle DT/ds}.

• In general Riemannian geometry, the derivative is computed using the Levi-Civita connection {displaystyle {bar {nabla }}} of the ambient manifold: {displaystyle DT/ds={bar {nabla }}_{T}T}. It splits into a tangent part and a normal part to the submanifold: {displaystyle {bar {nabla }}_{T}T=nabla _{T}T+({bar {nabla }}_{T}T)^{perp }}. The tangent part is the usual derivative {displaystyle nabla _{T}T} in {displaystyle M} (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is {displaystyle mathrm {I!I} (T,T)}, where {displaystyle mathrm {I!I} } denotes the second fundamental form.

• The Gauss–Bonnet theorem.