测地曲率 ：设P是曲线（C）上一点，
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
ε
=
n
×
α
{\displaystyle \varepsilon =n\times \alpha }
曲线（C）在P点的曲率向量
r
¨
=
k
β
{\displaystyle {\ddot {r}}=k\beta }
ε
{\displaystyle \varepsilon }
k
g
=
r
¨
⋅
ε
{\displaystyle k_{g}={\ddot {r}}\cdot \varepsilon }
称为曲线（C）在P点的测地曲率。
曲面S上的曲线（C），它在P点的测地曲率的绝对值等于（C）在P点的切平面上的正投影曲线（C'）的曲率。
k
2
=
k
g
2
+
k
n
2
{\displaystyle k^{2}=k_{g}^{2}+k_{n}^{2}}
式中，k为曲线在P点的曲率，
k
n
{\displaystyle k_{n}}
法曲率 。
今有一紧致 定向的二维曲面S，其线元素 可用曲面 的第一基本形式 的系数表示为：
d
s
2
=
E
d
u
2
+
2
F
d
u
d
v
+
G
d
v
2
{\displaystyle ds^{2}=Edu^{2}+2Fdudv+Gdv^{2}\,}
度量张量 可表成下列关系式：
(
g
i
j
)
=
(
g
11
g
12
g
21
g
22
)
=
(
E
F
F
G
)
{\displaystyle (g_{ij})={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}E&F\\F&G\\\end{pmatrix}}}
每当进行涉及到微分几何的实用演算时，都会用到其分量形式以利细部计算，因此有必要将前述向量形式定义的测地曲率以其分量形式来表征，以下将界定在二维曲面上局部范围，有关公式及其推导过程，可于列出的相关参考文献中找到。
Beltrami 公式令
C
{\displaystyle C}
s
{\displaystyle s}
C
{\displaystyle C}
C
:
r
(
s
)
=
(
u
(
s
)
,
v
(
s
)
)
{\displaystyle C:r(s)=(u(s),v(s))}
k
g
{\displaystyle k_{g}}
克氏符号 （全称克里斯多福符号 ，Christoffel symbols ）相关的表示式[ 1] [ 2] [ 3]
k
g
=
E
G
−
F
2
[
Γ
11
2
(
d
u
d
s
)
3
+
(
2
Γ
12
2
−
Γ
11
1
)
(
d
u
d
s
)
2
d
v
d
s
+
(
Γ
22
2
−
2
Γ
12
1
)
d
u
d
s
(
d
v
d
s
)
2
−
Γ
22
1
(
d
v
d
s
)
3
+
d
u
d
s
d
2
v
d
s
2
−
d
2
u
d
s
2
d
v
d
s
]
{\displaystyle k_{g}={\sqrt {EG-F^{2}}}\left[\Gamma _{11}^{2}\left({\frac {du}{ds}}\right)^{3}+\left(2\Gamma _{12}^{2}-\Gamma _{11}^{1}\right)\left({\frac {du}{ds}}\right)^{2}{\frac {dv}{ds}}+\left(\Gamma _{22}^{2}-2\Gamma _{12}^{1}\right){\frac {du}{ds}}\left({\frac {dv}{ds}}\right)^{2}-\Gamma _{22}^{1}\left({\frac {dv}{ds}}\right)^{3}+{\frac {du}{ds}}{\frac {d^{2}v}{ds^{2}}}-{\frac {d^{2}u}{ds^{2}}}{\frac {dv}{ds}}\right]}
上述用克式符号表示测地曲率的一般公式即是所谓的Beltrami 公式(Beltrami's formula for geodesic curvature.)[ 4] 克氏符号 Γk ij { k ij } 克式符号 属曲面的内蕴性质，而上述测地曲率 一般公式只和克式符号 与曲面 第一基本形式 有关，因此，测地曲率 必然是属曲面的内蕴几何 量[ 5]
今若曲线
C
{\displaystyle C}
u
=
(
s
)
{\displaystyle u=(s)}
v
=
{\displaystyle v=}
d
v
/
d
s
=
0
{\displaystyle dv/ds=0}
d
u
/
d
s
=
1
/
g
11
{\displaystyle du/ds=1/{\sqrt {g_{11}}}}
(
k
g
)
u
−
l
i
n
e
=
Γ
11
2
E
G
−
F
2
E
E
=
Γ
11
2
(
g
1
/
2
g
11
3
/
2
)
{\displaystyle (k_{g})_{u-line}=\Gamma _{11}^{2}{\dfrac {\sqrt {EG-F^{2}}}{E{\sqrt {E}}}}=\Gamma _{11}^{2}\left({\dfrac {g^{1/2}}{g_{11}^{3/2}}}\right)}
同理，假如曲线
C
{\displaystyle C}
v
=
(
s
)
{\displaystyle v=(s)}
u
=
{\displaystyle u=}
d
u
/
d
s
=
0
{\displaystyle du/ds=0}
d
v
/
d
s
=
1
/
g
22
{\displaystyle dv/ds=1/{\sqrt {g_{22}}}}
(
k
g
)
v
−
l
i
n
e
=
−
Γ
22
1
E
G
−
F
2
G
G
=
−
Γ
22
1
(
g
1
/
2
g
22
3
/
2
)
{\displaystyle (k_{g})_{v-line}=-\Gamma _{22}^{1}{\dfrac {\sqrt {EG-F^{2}}}{G{\sqrt {G}}}}=-\Gamma _{22}^{1}\left({\dfrac {g^{1/2}}{g_{22}^{3/2}}}\right)}
Liouville 公式令
C
{\displaystyle C}
s
{\displaystyle s}
C
{\displaystyle C}
C
:
r
(
s
)
=
(
u
(
s
)
,
v
(
s
)
)
{\displaystyle C:r(s)=(u(s),v(s))}
正交座标系 ，换言之，第一基本形式 的系数
F
=
0
{\displaystyle F=0}
C
{\displaystyle C}
u
{\displaystyle u}
θ
{\displaystyle \theta }
k
g
{\displaystyle k_{g}}
θ
(
s
)
{\displaystyle \theta (s)}
Liouville 公式[ 6] [ 7] [ 8]
k
g
=
d
θ
(
s
)
d
s
−
1
2
G
∂
ln
E
∂
v
cos
θ
+
1
2
E
∂
ln
G
∂
u
sin
θ
=
d
θ
(
s
)
d
s
+
(
k
g
)
u
−
l
i
n
e
cos
θ
+
(
k
g
)
v
−
l
i
n
e
sin
θ
=
d
θ
(
s
)
d
s
+
(
k
g
)
u
−
l
i
n
e
E
d
u
d
s
+
(
k
g
)
v
−
l
i
n
e
G
d
v
d
s
{\displaystyle {\begin{aligned}k_{g}&={\dfrac {d\theta (s)}{ds}}-{\dfrac {1}{2{\sqrt {G}}}}{\dfrac {\partial \ln E}{\partial v}}\cos \theta +{\dfrac {1}{2{\sqrt {E}}}}{\dfrac {\partial \ln G}{\partial u}}\sin \theta \\&={\dfrac {d\theta (s)}{ds}}+(k_{g})_{u-line}\cos \theta +(k_{g})_{v-line}\sin \theta \\&={\dfrac {d\theta (s)}{ds}}+(k_{g})_{u-line}{\sqrt {E}}{\dfrac {du}{ds}}+(k_{g})_{v-line}{\sqrt {G}}{\dfrac {dv}{ds}}\end{aligned}}}
(
k
g
)
u
−
l
i
n
e
{\displaystyle (k_{g})_{u-line}}
(
k
g
)
v
−
l
i
n
e
{\displaystyle (k_{g})_{v-line}}
C
{\displaystyle C}
u
=
(
s
)
{\displaystyle u=(s)}
v
=
{\displaystyle v=}
d
v
/
d
s
=
0
{\displaystyle dv/ds=0}
d
u
/
d
s
=
1
/
E
{\displaystyle du/ds=1/{\sqrt {E}}}
(
k
g
)
u
−
l
i
n
e
=
−
E
v
2
E
G
{\displaystyle (k_{g})_{u-line}=-{\dfrac {E_{v}}{2E{\sqrt {G}}}}}
同理，假如曲线
C
{\displaystyle C}
v
=
(
s
)
{\displaystyle v=(s)}
u
=
{\displaystyle u=}
d
u
/
d
s
=
0
{\displaystyle du/ds=0}
d
v
/
d
s
=
1
/
G
{\displaystyle dv/ds=1/{\sqrt {G}}}
(
k
g
)
v
−
l
i
n
e
=
G
u
2
G
E
{\displaystyle (k_{g})_{v-line}={\dfrac {G_{u}}{2G{\sqrt {E}}}}}
以上测地曲率之Liouville公式就已列出有三种，若觉得怎么会有这么多样形式，其实还有其他变形，例如可参考网络上更加精简且优美的形式[ 9]
^ Kreyszig, Erwin. Differential Geometry . Dover Publications, New York. 1991: 154 -156. ISBN 978-0-486-66721-8 ^ Patrikalakis, Nicholas M.; Maekawa, Takashi. Shape Interrogation for Computer Aided Design and Manufacturing . Springer, New York. 2002: 266 -268. ISBN 978-3-642-04073-3 【推导过程见MIT线上开放课程 §10.2.1. Parametric surfaces】 （页面存档备份 ，存于互联网档案馆 ）^ Blaga, Paul A. Lectures on the Differential Geometry of Curves and Surfaces . Napoca Press, Cluj-Napoca, Romania. 2005: 177 -179. ISBN 9736568962 ^ Nayak, Prasun Kumar. Textbook of Tensor Calculus and Differential Geometry. PHI Learning Pvt. Ltd., New Delhi. 2011: 364,369. ^ Slobodyan, Yu.S., Geodesic curvature , Hazewinkel, Michiel (编), 数学百科全书 , Springer , 1989, ISBN 978-1-55608-010-4 ^ do Carmo, Manfredo P. Differential Geometry of Curves and Surfaces. Prentice-Hall. 1976: 253-254. ISBN 0-13-212589-7 ^ Gray, Alfred; Abbena, Elsa; Salamon, Simon. Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition. Chapman & Hall/CRC. 2006: 904-905. ISBN 978-1584884484 ^ Dube, K.K. Differential Geometry and Tensors. I. K. International Pvt Ltd. 2009: 200-201. ISBN 978-9380026589 ^ Sigurd Angenent. A note and two problems on Liouville's formula. （页面存档备份 ，存于互联网档案馆 ） 这是介绍测地曲率之Liouville公式更加精简形式的文件。
![{\displaystyle k_{g}={\sqrt {EG-F^{2}}}\left[\Gamma _{11}^{2}\left({\frac {du}{ds}}\right)^{3}+\left(2\Gamma _{12}^{2}-\Gamma _{11}^{1}\right)\left({\frac {du}{ds}}\right)^{2}{\frac {dv}{ds}}+\left(\Gamma _{22}^{2}-2\Gamma _{12}^{1}\right){\frac {du}{ds}}\left({\frac {dv}{ds}}\right)^{2}-\Gamma _{22}^{1}\left({\frac {dv}{ds}}\right)^{3}+{\frac {du}{ds}}{\frac {d^{2}v}{ds^{2}}}-{\frac {d^{2}u}{ds^{2}}}{\frac {dv}{ds}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da825341ad4926184669e0cecad12cd6c0b9b610)
