In Riemannian geometry, the geodesic curvature
k
g
of a curve
γ
measures how far the curve is from being a geodesic. In a given manifold
M
¯
, the geodesic curvature is just the usual curvature of
γ
(see below), but when
γ
is restricted to lie on a submanifold
M
of
M
¯
(e.g. for curves on surfaces), geodesic curvature refers to the curvature of
γ
in
M
and it is different in general from the curvature of
γ
in the ambient manifold
M
¯
. The (ambient) curvature
k
of
γ
depends on two factors: the curvature of the submanifold
M
in the direction of
γ
(the normal curvature
k
n
), which depends only on the direction of the curve, and the curvature of
γ
seen in
M
(the geodesic curvature
k
g
), which is a second order quantity. The relation between these is
k
=
k
g
2
+
k
n
2
. In particular geodesics on
M
have zero geodesic curvature (they are "straight"), so that
k
=
k
n
, which explains why they appear to be curved in ambient space whenever the submanifold is.
Consider a curve
γ
in a manifold
M
¯
, parametrized by arclength, with unit tangent vector
T
=
d
γ
/
d
s
. Its curvature is the norm of the covariant derivative of
T
:
k
=
∥
D
T
/
d
s
∥
. If
γ
lies on
M
, the geodesic curvature is the norm of the projection of the covariant derivative
D
T
/
d
s
on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of
D
T
/
d
s
on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space
R
n
, then the covariant derivative
D
T
/
d
s
is just the usual derivative
d
T
/
d
s
.
Let
M
be the unit sphere
S
2
in three-dimensional Euclidean space. The normal curvature of
S
2
is identically 1, independently of the direction considered. Great circles have curvature
k
=
1
, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius
r
will have curvature
1
/
r
and geodesic curvature
k
g
=
1
−
r
2
r
.
The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold
M
. It does not depend on the way the submanifold
M
sits in
M
¯
.
Geodesics of
M
have zero geodesic curvature, which is equivalent to saying that
D
T
/
d
s
is orthogonal to the tangent space to
M
.
On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve:
k
n
only depends on the point on the submanifold and the direction
T
, but not on
D
T
/
d
s
.
In general Riemannian geometry, the derivative is computed using the Levi-Civita connection
∇
¯
of the ambient manifold:
D
T
/
d
s
=
∇
¯
T
T
. It splits into a tangent part and a normal part to the submanifold:
∇
¯
T
T
=
∇
T
T
+
(
∇
¯
T
T
)
⊥
. The tangent part is the usual derivative
∇
T
T
in
M
(it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is
I
I
(
T
,
T
)
, where
I
I
denotes the second fundamental form.
The Gauss–Bonnet theorem.
