- Source: Geodesic curvature
In Riemannian geometry, the geodesic curvature
k
g
{\displaystyle k_{g}}
of a curve
γ
{\displaystyle \gamma }
measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold
M
¯
{\displaystyle {\bar {M}}}
, the geodesic curvature is just the usual curvature of
γ
{\displaystyle \gamma }
(see below). However, when the curve
γ
{\displaystyle \gamma }
is restricted to lie on a submanifold
M
{\displaystyle M}
of
M
¯
{\displaystyle {\bar {M}}}
(e.g. for curves on surfaces), geodesic curvature refers to the curvature of
γ
{\displaystyle \gamma }
in
M
{\displaystyle M}
and it is different in general from the curvature of
γ
{\displaystyle \gamma }
in the ambient manifold
M
¯
{\displaystyle {\bar {M}}}
. The (ambient) curvature
k
{\displaystyle k}
of
γ
{\displaystyle \gamma }
depends on two factors: the curvature of the submanifold
M
{\displaystyle M}
in the direction of
γ
{\displaystyle \gamma }
(the normal curvature
k
n
{\displaystyle k_{n}}
), which depends only on the direction of the curve, and the curvature of
γ
{\displaystyle \gamma }
seen in
M
{\displaystyle M}
(the geodesic curvature
k
g
{\displaystyle k_{g}}
), which is a second order quantity. The relation between these is
k
=
k
g
2
+
k
n
2
{\displaystyle k={\sqrt {k_{g}^{2}+k_{n}^{2}}}}
. In particular geodesics on
M
{\displaystyle M}
have zero geodesic curvature (they are "straight"), so that
k
=
k
n
{\displaystyle k=k_{n}}
, which explains why they appear to be curved in ambient space whenever the submanifold is.
Definition
Consider a curve
γ
{\displaystyle \gamma }
in a manifold
M
¯
{\displaystyle {\bar {M}}}
, parametrized by arclength, with unit tangent vector
T
=
d
γ
/
d
s
{\displaystyle T=d\gamma /ds}
. Its curvature is the norm of the covariant derivative of
T
{\displaystyle T}
:
k
=
‖
D
T
/
d
s
‖
{\displaystyle k=\|DT/ds\|}
. If
γ
{\displaystyle \gamma }
lies on
M
{\displaystyle M}
, the geodesic curvature is the norm of the projection of the covariant derivative
D
T
/
d
s
{\displaystyle DT/ds}
on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of
D
T
/
d
s
{\displaystyle DT/ds}
on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
, then the covariant derivative
D
T
/
d
s
{\displaystyle DT/ds}
is just the usual derivative
d
T
/
d
s
{\displaystyle dT/ds}
.
If
γ
{\displaystyle \gamma }
is unit-speed, i.e.
‖
γ
′
(
s
)
‖
=
1
{\displaystyle \|\gamma '(s)\|=1}
, and
N
{\displaystyle N}
designates the unit normal field of
M
{\displaystyle M}
along
γ
{\displaystyle \gamma }
, the geodesic curvature is given by
k
g
=
γ
″
(
s
)
⋅
(
N
(
γ
(
s
)
)
×
γ
′
(
s
)
)
=
[
d
2
γ
(
s
)
d
s
2
,
N
(
γ
(
s
)
)
,
d
γ
(
s
)
d
s
]
,
{\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.
Example
Let
M
{\displaystyle M}
be the unit sphere
S
2
{\displaystyle S^{2}}
in three-dimensional Euclidean space. The normal curvature of
S
2
{\displaystyle S^{2}}
is identically 1, independently of the direction considered. Great circles have curvature
k
=
1
{\displaystyle k=1}
, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius
r
{\displaystyle r}
will have curvature
1
/
r
{\displaystyle 1/r}
and geodesic curvature
k
g
=
1
−
r
2
r
{\displaystyle k_{g}={\frac {\sqrt {1-r^{2}}}{r}}}
.
Some results involving geodesic curvature
The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold
M
{\displaystyle M}
. It does not depend on the way the submanifold
M
{\displaystyle M}
sits in
M
¯
{\displaystyle {\bar {M}}}
.
Geodesics of
M
{\displaystyle M}
have zero geodesic curvature, which is equivalent to saying that
D
T
/
d
s
{\displaystyle DT/ds}
is orthogonal to the tangent space to
M
{\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:
k
n
{\displaystyle k_{n}}
only depends on the point on the submanifold and the direction
T
{\displaystyle T}
, but not on
D
T
/
d
s
{\displaystyle DT/ds}
.
In general Riemannian geometry, the derivative is computed using the Levi-Civita connection
∇
¯
{\displaystyle {\bar {\nabla }}}
of the ambient manifold:
D
T
/
d
s
=
∇
¯
T
T
{\displaystyle DT/ds={\bar {\nabla }}_{T}T}
. It splits into a tangent part and a normal part to the submanifold:
∇
¯
T
T
=
∇
T
T
+
(
∇
¯
T
T
)
⊥
{\displaystyle {\bar {\nabla }}_{T}T=\nabla _{T}T+({\bar {\nabla }}_{T}T)^{\perp }}
. The tangent part is the usual derivative
∇
T
T
{\displaystyle \nabla _{T}T}
in
M
{\displaystyle M}
(it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is
I
I
(
T
,
T
)
{\displaystyle \mathrm {I\!I} (T,T)}
, where
I
I
{\displaystyle \mathrm {I\!I} }
denotes the second fundamental form.
The Gauss–Bonnet theorem.
See also
Curvature
Darboux frame
Gauss–Codazzi equations
References
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.
External links
Weisstein, Eric W. "Geodesic curvature". MathWorld.
Kata Kunci Pencarian:
- Teorema Gauss–Bonnet
- Teorema ketunggalan Alexandrov
- Daftar masalah matematika yang belum terpecahkan
- Geodesic curvature
- Geodesic
- Differential geometry of surfaces
- Curvature
- Gauss–Bonnet theorem
- Ricci curvature
- Riemann curvature tensor
- Sectional curvature
- Scalar curvature
- Geodesics in general relativity
