Consider a curve ${\displaystyle \gamma }$  in a manifold ${\displaystyle {\bar {M}}}$ , parametrized by arclength, with unit tangent vector ${\displaystyle T=d\gamma /ds}$ . 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 ${\displaystyle dT/ds}$ .

If ${\displaystyle \gamma }$  is unit-speed, i.e. ${\displaystyle \|\gamma '(s)\|=1}$ , and ${\displaystyle N}$  designates the unit normal field of ${\displaystyle M}$  along ${\displaystyle \gamma }$ , the geodesic curvature is given by

${\displaystyle k_{g}=\gamma ''(s)\cdot {\Big (}N(\gamma (s))\times \gamma '(s){\Big )}=\left[{\frac {\mathrm {d} ^{2}\gamma (s)}{\mathrm {d} s^{2}}},N(\gamma (s)),{\frac {\mathrm {d} \gamma (s)}{\mathrm {d} s}}\right]\,,}$ 

where the square brackets denote the scalar triple product.

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.

• Curvature
• Darboux frame
• Gauss–Codazzi equations

• do Carmo, Manfredo P. (1976), Differential Geometry of Curves and Surfaces, Prentice-Hall, ISBN 0-13-212589-7
• Guggenheimer, Heinrich (1977), "Surfaces", Differential Geometry, Dover, ISBN 0-486-63433-7.
• Slobodyan, Yu.S. (2001) [1994], "Geodesic curvature", Encyclopedia of Mathematics, EMS Press.

• Weisstein, Eric W. "Geodesic curvature". MathWorld.