Isometries; Conformal Maps

Definition A diffeomorphism ϕ : S → ¯S is an isometryif

hw_{1}, w_{2}i_{p} = hdϕ_{p}(w_{1}), dϕ_{p}(w_{2})i_{ϕ(p)} ∀ p ∈ S, ∀ w_{1}, w_{2} ∈ T_{p}S.

The surfaces S and ¯S are then said to beisometric.

In other words, a diffeomorphism ϕ is an isometry if the differential dϕ preserves the inner
product. It follows that, dϕ_{p} : T_{p}S → T_{ϕ(p)}S being an isometry,¯

I_{p}(w) = hw, wi_{p} = hdϕ(w), dϕ(w)i_{ϕ(p)}= I_{ϕ(p)}(dϕ_{p}(w)) ∀ w ∈ T_{p}S,
i.e. the diffeomorphism ϕ : S → ¯S prserves the first fundamental form.

Conversely, if the diffeomorphism ϕ : S → ¯S prserves the first fundamental form, then
2hw_{1}, w_{2}i = I_{p}(w_{1}+ w_{2}) − I_{p}(w_{1}) − I_{p}(w_{2}) ∀ w_{1}, w_{2} ∈ T_{p}S

= I_{ϕ(p)}(dϕ_{p}(w_{1}+ w_{2})) − I_{ϕ(p)}(dϕ_{p}(w_{1})) − I_{ϕ(p)}(dϕ_{p}(w_{2}))

= hdϕ(w_{1}), dϕ(w_{2})i,
and ϕ is, therefore, an isometry.

Definition A map ϕ : V → ¯S of a neighborhood V ⊂ S of p ∈ S is alocal isometry at p if there exists a neighborhood ¯V ⊂ ¯S of ϕ(p) ∈ ¯S such that ϕ : V → ¯V is an isometry. If there exists a local isometry into ¯S at every p ∈ S, the surface S is said to be locally isometric to ¯S.

It is clear that if ϕ : S → ¯S is a diffeomorphism and a local isometry for every p ∈ S, then ϕ is an isometry (globally).

However, a local isometry is not necessary an isometry globally, e.g. the xy-plane P = {(x, y, z) ∈
R^{3} | z = 0} and the cylinder S = {(x, y, z) ∈ R^{3} | x^{2}+ y^{2} = 1} are locally isometric, but they
are not homeomorphic, so P and S are not diffeomorphic or isometric globally.

Since any simple closed curve C ⊂ P in the plane P can be shrunk (deformed) continuously into a point without leaving the plane P, and this topological property in P is preserved by a homeomorphism ϕ : P → ϕ(P ).

Note that a parallel C^{0}, e.g. C^{0} = {(cos u, sin u, 0) | u ∈ [0, 2π]} ⊂ S, of the cylinder S does not
have that property while the corresponding unit circle C = {(x, y, 0) | x^{2}+ y^{2} = 1} in P can be
shrunk continuously into a point without leaving the plane P, so P and S are not homeomorphic.

Proposition Suppose that there exist parametrizations X : U → S and ¯X : U → S such that
E = ¯E, F = ¯F , G = ¯G in U. Then the map ϕ = ¯X ◦ X^{−1} : X(U ) → ¯S is a local isometry.

Proof Let p ∈ X(U ) and w ∈ T_{p}S. Then w is tangent to a curve X(α(t)) at t = 0, where
α(t) = (u(t), v(t)) is a curve in U ; thus, w may be written (t = 0)

w = X_{u}u^{0}+ X_{v}v^{0}.

By definition, the vector dϕ_{p}(w) is the tangent vector to the curve ¯X ◦ X^{−1}◦ X(α(t)) = ¯X(α(t))
at t = 0. Thus,

dϕ_{p}(w) = ¯X_{u}u^{0} + ¯X_{v}v^{0}.
Since

Ip(w) = E(u^{0})^{2}+ 2F u^{0}v^{0}+ G(v^{0})^{2},
I_{ϕ(p)}(dϕ_{p}(w)) = E(u¯ ^{0})^{2}+ 2 ¯F u^{0}v^{0}+ ¯G(v^{0})^{2},

and the assumption E = ¯E, F = ¯F , G = ¯G in U, we conclude that I_{p}(w) = I_{ϕ(p)}(dϕ_{p}(w)) for all
p ∈ X(U ) and all w ∈ TpS; hence, ϕ is a local isometry.

Definition A diffeomorphism ϕ : S → ¯S is called a conformal mapif

hdϕ_{p}(w_{1}), dϕ_{p}(w_{2})i = λ^{2}(p) hw_{1}, w_{2}i^{w}=⇒ |dϕ^{1}^{=w}^{2} _{p}(w_{1})|^{2} = λ^{2}(p) |w_{1}|^{2} ∀ p ∈ S, ∀ w_{1}, w_{2} ∈ T_{p}S,
where λ^{2} is a nowhere-zero differentiable function on S; the surface S and ¯S are then said to be
conformal. A map ϕ : V → ¯S of a neighborhood V ⊂ S of p ∈ S is a local conformal map at
p if there exists a neighborhood ¯V ⊂ ¯S of ϕ(p) ∈ ¯S such that ϕ : V → ¯V is a conformal map.

If there exists a local conformal map into ¯S at every p ∈ S, the surface S is said to be locally conformalto ¯S.

The geometric meaning of the above definition is thatthe angles (but not necessarily the lengths) are preserved by conformal maps. In fact, let α : I → S and β : I → S be two curves in S which intersect at, say, t = 0. Their angle θ at t = 0 is given by

cos θ = hα^{0}, β^{0}i

|α^{0}| |β^{0}|, 0 ≤ θ ≤ π.

A conformal map ϕ : S → ¯S maps these curves into ϕ ◦ α : I → ¯S, ϕ ◦ β : I → ¯S, which intersect when t = 0, making an angle ¯θ given by

cos ¯θ = hdϕ(α^{0}), dϕ(β^{0})i

|dϕ(α^{0})| |dϕ(β^{0})| = λ^{2}hα^{0}, β^{0}i

λ^{2}|α^{0}| |β^{0}| = cos θ.

Proposition Suppose that there exist parametrizations X : U → S and ¯X : U → ¯S such that
E = λ^{2}E, F = λ¯ ^{2}F , G = λ¯ ^{2}G in U, where λ¯ ^{2} is a nowhere-zero differentiable function in U. Then
the map ϕ = ¯X ◦ X^{−1} : X(U ) → ¯S is a local conformal map.

Example For a > 0, let

X(u, v) = (a cosh v cos u, a cosh v sin u, av), (u, v) ∈ U = {0 < u < 2π, −∞ < v < ∞}

X(¯¯ u, ¯v) = (¯v cos ¯u, ¯v sin ¯u, a¯u), (¯u, ¯v) ∈ U = {0 < ¯u < 2π, −∞ < ¯v < ∞}

be parametrizations of the catenoid S and the helicoid ¯S, rspectively. Then the coefficients of the first fundamental forms are

E = a^{2}cosh^{2}v, F = 0, G = a^{2}(1 + sinh^{2}v) = a^{2}cosh^{2}v ∀ (u, v) ∈ U,
E = a¯ ^{2}+ ¯v^{2}, F = 0,¯ G = 1¯ ∀ (¯u, ¯v) ∈ U.

Let us make the following change of parameters

¯

u = u, ¯v = a sinh v, ∀ (u, v) ∈ U, which is possible since the map is clearly one-to-one, and the Jacobian

∂(¯u, ¯v)

∂(u, v) = a cosh v 6= 0 ∀ (u, v) ∈ U.

Thus,

X(u, v) = (a sinh v cos u, a sinh v sin u, au),¯ (u, v) ∈ U, is a new parametrization of the helicoid with

E = a^{2}cosh^{2}v, F = 0, G = a^{2}cosh^{2}v ∀ (u, v) ∈ U.

We conclude that the catenoid and the helicoid are locally isometric.

Example Let S be the one-sheeted cone (minus the vertex) z = kp

x^{2}+ y^{2}, k > 0, (x, y) 6= (0, 0),
and let U ⊂ R^{2} be the open set given in polar coordinates (ρ, θ) by

0 < ρ < ∞, 0 < θ < 2π sin α,

where 2α (0 < 2α < π) is the angle at the vertex of the cone (i.e., where cot α = k), and let
F : U → S ⊂ R^{3} be the map

F (ρ, θ) =

ρ sin α cos

θ sin α

, ρ sin α sin

θ sin α

, ρ cos α

. Then

• F (U ) ⊂ S, since

kp

x^{2}+ y^{2} = cot α
q

ρ^{2}sin^{2}α = ρ cos α = z,

• F : U → S \ {(ρ sin α, 0, ρ cos α) | 0 < ρ < ∞} is a diffeomorphism from U onto the cone minus a generator θ = 0, since F and dF are one-to-one in U,

and thus F (ρ, θ) is a parametrization of S with the coefficients of the first fundamental form being

E = hF_{ρ}, F_{ρ}i = 1, F = hF_{ρ}, F_{θ}i = 0, G = hF_{θ}, F_{θ}i = ρ^{2},
Also since U may be viewed as a regular surface parametrized by

X(ρ, θ) = (ρ cos θ, ρ sin θ, 0) ∈ R¯ ^{3}, 0 < ρ < ∞, 0 < θ < 2π sin α,
with the coefficients of the first fundamental form of U in this parametrization being

E = h ¯¯ X_{ρ}, ¯X_{ρ}i = 1 = E, F = h ¯¯ X_{ρ}, ¯X_{θ}i = 0 = F, G = h ¯¯ X_{θ}, ¯X_{θ}i = ρ^{2} = G,
F : U → S is a local isometry.

The most important property of conformal maps is given by the following theorem, which we shall not prove.

Theorem Any two regular surfaces are locally conformal.

The proof is based on the possibility of parametrizing a neighborhood of any point of a regular surface in such a way that the coefficients of the first fundamental form are

E = λ^{2}(u, v), F = 0, G = λ^{2}(u, v).

Such a coordinate system is called isothermal. Once the existence of an isothermal coordinate system of a regular surface S is assumed, S is clearly locally conformal to a plane, and by composition locally conformal to any other surface.

The Gauss Theorem and the Equations of Compatibility

Let X : U ⊂ R^{2} → S be a parametrization in the orientation of S. At each p ∈ X(U ), since
X_{u}, X_{v}, N ∈ R^{3} are linearly independent, we may express vectors X_{uu}, X_{uv}, X_{vu}, X_{vv}, N_{u}, N_{v} ∈
R^{3} in the basis {Xu, Xv, N } and obtain

X_{uu} = Γ^{1}_{11}X_{u}+ Γ^{2}_{11}X_{v}+ eN
X_{uv} = Γ^{1}_{12}X_{u}+ Γ^{2}_{12}X_{v}+ f N
X_{vu} = Γ^{1}_{21}X_{u}+ Γ^{2}_{21}X_{v}+ f N
Xvv = Γ^{1}_{22}Xu+ Γ^{2}_{22}Xv+ gN

N_{u} = a_{11}X_{u} + a_{21}X_{v}
N_{v} = a_{12}X_{u} + a_{22}X_{v}

where the aij, i, j = 1, 2, were obtained in Chapter 3 and the coefficients Γ^{k}_{ij}, i, j = 1, 2, are
called the Christoffel symbols of S in the parametrization X. Since X_{uv}= X_{vu}, we conclude that
Γ^{1}_{12} = Γ^{1}_{21} and Γ^{2}_{12} = Γ^{2}_{21}; that is, the Christoffel symbols are symmetric relative to the lower
indices.

To determine the Christoffel symbols, we take the inner product of the first four relations with
X_{u} and X_{v}, obtaining the system

E F

F G

Γ^{1}_{11}
Γ^{2}_{11}

=

hX_{uu}, X_{u}i
hX_{uu}, X_{v}i

=

1 2Eu

F_{u}− 1
2E_{v}

=⇒

Γ^{1}_{11}
Γ^{2}_{11}

=

E F

F G

−1

1 2Eu

F_{u}− 1
2E_{v}

Γ^{1}_{12}
Γ^{2}_{12}

=

hX_{uv}, X_{u}i
hX_{uv}, X_{v}i

=

1
2E_{v}
1
2G_{u}

=⇒

Γ^{1}_{21}
Γ^{2}_{21}

=

Γ^{1}_{12}
Γ^{2}_{12}

=

E F

F G

−1

1
2E_{v}
1
2G_{u}

E F

F G

Γ^{1}_{22}
Γ^{2}_{22}

=

hX_{vv}, X_{u}i
hX_{vv}, X_{v}i

=

F_{v}− 1
2G_{u}
1
2G_{v}

=⇒

Γ^{1}_{22}
Γ^{2}_{22}

=

E F

F G

−1

F_{v}− 1
2G_{u}
1
2G_{v}

where we have used hXuu, Xui = 1

2

∂

∂uhXu, Xui = 1

2Eu, hXuu, Xvi = ∂

∂uhXu, Xvi−hXu, Xvui = Fu−hXu, Xuvi = Fu−1 2Ev

In particular, if X is an orthogonal parametrization, i.e. F = hX_{u}, X_{v}i = 0 at each p ∈ X(U ),
then

Γ^{1}_{11}
Γ^{2}_{11}

= 1

2(EG − F^{2})

GE_{u}

−EE_{v}

Γ^{1}_{12}
Γ^{2}_{12}

= 1

2(EG − F^{2})

GE_{v}
EG_{u}

Γ^{1}_{22}
Γ^{2}_{22}

= 1

2(EG − F^{2})

GG_{u}
EG_{v}

Thus, it is possible to solve the above system and to compute the Christoffel symbols in terms of the coefficients of the first fundamental form, E, F, G and their derivatives. Hence, all geo- metric concepts and properties expressed in terms ofthe Christoffel symbols are invariant under isometries.

Example Let S be a surface of revolution parametrized by

X(u, v) = (f (v) cos u, f (v) sin v, g(v)), f (v) 6= 0.

Since

E = (f (v))^{2}, F = 0, G = (f^{0}(v))^{2}+ (g^{0}(v))^{2},
we obtain

E_{u} = 0, E_{v} = 2f f^{0}, F_{u} = F_{v} = 0, G_{u} = 0, G_{v} = 2(f^{0}f^{00}+ g^{0}g^{00}),
and

Γ^{1}_{11}= 0, Γ^{2}_{11} = − f f^{0}

(f^{0})^{2} + (g^{0})^{2}, Γ^{1}_{12}= f f^{0}

f^{2} , Γ^{2}_{12}= 0, Γ^{1}_{22} = 0, Γ^{2}_{22}= f^{0}f^{00}+ g^{0}g^{00}
(f^{0})^{2}+ (g^{0})^{2}.
Since X : U ⊂ R^{2} → R^{3} is differentiable,

(X_{uu})_{v} = (X_{uv})_{u},

⇐⇒ Γ^{1}_{11}X_{u}+ Γ^{2}_{11}X_{v}+ eN

v = Γ^{1}_{12}X_{u}+ Γ^{2}_{12}X_{v}+ f N

u

⇐⇒ Γ^{1}_{11}X_{uv}+ Γ^{2}_{11}X_{vv}+ eN_{v}+ (Γ^{1}_{11})_{v}X_{u}+ (Γ^{2}_{11})_{v}X_{v}+ e_{v}N

=Γ^{1}_{12}X_{uu}+ Γ^{2}_{12}X_{vu}+ f N_{u}+ (Γ^{1}_{12})_{u}X_{u}+ (Γ^{2}_{12})_{u}X_{v} + f_{u}N (∗)
By equating the coefficients of X_{v}, and using

a_{11} a_{21}
a12 a22

= − e f f g

E F

F G

−1

= −1

EG − F^{2}

eG − f F −eF + f E f G − gF −f F + gE

. we obtain the following formula for the Gaussian curvature K

Γ^{1}_{11}Γ^{2}_{12}+ Γ^{2}_{11}Γ^{2}_{22}+ ea22+ (Γ^{2}_{11})v = Γ^{1}_{12}Γ^{2}_{11}+ Γ^{2}_{12}Γ^{2}_{12}+ f a21+ (Γ^{2}_{12})u

⇐⇒ (Γ^{2}_{12})_{u}− (Γ^{2}_{11})_{v}+ Γ^{1}_{12}Γ^{2}_{11}+ Γ^{2}_{12}Γ^{2}_{12}− Γ^{1}_{11}Γ^{2}_{12}− Γ^{2}_{11}Γ^{2}_{22}= ea_{22}− f a_{21}

⇐⇒ (Γ^{2}_{12})u− (Γ^{2}_{11})v+ Γ^{1}_{12}Γ^{2}_{11}+ Γ^{2}_{12}Γ^{2}_{12}− Γ^{1}_{11}Γ^{2}_{12}− Γ^{2}_{11}Γ^{2}_{22}= −E eg − f^{2}
EG − F^{2}

⇐⇒ (Γ^{2}_{12})_{u}− (Γ^{2}_{11})_{v}+ Γ^{1}_{12}Γ^{2}_{11}+ Γ^{2}_{12}Γ^{2}_{12}− Γ^{1}_{11}Γ^{2}_{12}− Γ^{2}_{11}Γ^{2}_{22}= −EK,

THEOREMA EGREGIUM (Gauss) The Gaussian curvature K of a surface is invariant by local isometries.

Remarks

• By equating the coefficients of X_{u}in equation(∗), we obtain another formula of the Gaussian
curvature K.

(Γ^{1}_{12})u− (Γ^{1}_{11})v+ Γ^{1}_{12}Γ^{1}_{11}+ Γ^{2}_{12}Γ^{1}_{12}− Γ^{1}_{11}Γ^{1}_{12}− Γ^{2}_{11}Γ^{1}_{22} = (Γ^{1}_{12})u− (Γ^{1}_{11})v + Γ^{2}_{12}Γ^{1}_{12}− Γ^{2}_{11}Γ^{1}_{22}= F K.

• By equating the coefficients of N in equation (∗), we obtain

e_{v} − f_{u} = eΓ^{1}_{12}+ f (Γ^{2}_{12}− Γ^{1}_{11}) − gΓ^{2}_{11} (†).

• By equating the coefficients of N in equation (X_{vv})_{u}− (X_{vu})_{v} = 0, we obtain
f_{v}− g_{u} = eΓ^{1}_{22}+ f (Γ^{2}_{22}− Γ^{1}_{12}) − gΓ^{2}_{12} (††).

Equations(†) and (††)are called Mainardi-Codazzi equations.

Theorem (Bonnet) Let E, F, G, e, f, g be differentiable functions, defined in an open set
V ⊂ R^{2}, with E > 0 and G > 0. Assume that

• the given functions satisfy formally the Gauss and Mainardi-Codazzi equations,

• and that EG − F^{2} > 0.

Then,

• for every q ∈ V there exists a neighborhood U ⊂ V of q,

• and a diffeomorphism X : U → X(U ) ⊂ R^{3}

such that the regular surface X(U ) ⊂ R^{3} has E, F, G and e, f, g as coefficients of the first and
second fundamental forms, respectively.

Furthermore, if U is connected and if

X : U → ¯¯ X(U ) ⊂ R^{3}

is another diffeomorphism satisfying the same conditions, then there exist a translation T and a
proper linear orthogonal transformation ρ in R^{3} such that

X = T ◦ ρ ◦ X.¯

Remark In the following, we shall calculate the Christoffel symbols and Gaussian curvatures in
terms of the metric tensor (g_{ij}) and its partial derivatives.

Let U be an open subset in the u_{1}u_{2}-plane, and X : U ⊂ R^{2} → S be a parametrization in the
orientation of S. At each p ∈ X(U ), let X1 = Xu1, X2 = Xu2, and let

g_{11} = hX_{1}, X_{1}i = E, g_{12} = g_{21}= hX_{1}, X_{2}i = F, g_{22} = hX_{2}, X_{2}i = G ⇐⇒ g_{11} g_{12}
g_{21} g_{22}

=E F

F G

, and

g^{11} g^{12}
g^{21} g^{22}

= (g^{ij}) = (g_{ij})^{−1} = 1
det(g_{ij})

g_{22} −g_{12}

−g_{21} g_{11}

= 1

EG − F^{2}

G −F

−F E

. Note that

2

X

k=1

g^{mk}g_{k`} ^{(†)}= δ_{m`} =

(1 if m = ` 0 if m 6= `.

Since X1, X2, N ∈ R^{3} are linearly independent, we may express vectors Xij = Xuiuj ∈ R^{3} and
N_{i} = N_{u}_{i} ∈ T_{p}S as

X_{ij} ^{(∗)}=

2

X

k=1

Γ^{k}_{ij}X_{k}+ h_{ij}N, i, j = 1, 2, where h_{11} h_{12}
h_{21} h_{22}

= e f f g

Ni (∗∗)=

2

X

j=1

ajiXj, i = 1, 2,

where

a_{11} a_{21}
a_{12} a_{22}

= −h_{11} h_{12}
h_{21} h_{22}

g^{11} g^{12}
g^{21} g^{22}

= 1

EG − F^{2}

h_{11} h_{12}
h_{21} h_{22}

−g_{22} g_{21}
g_{12} −g_{11}

,

as obtained in Chapter 3 and the coefficients Γ^{k}_{ij}, i, j = 1, 2, are called theChristoffel symbols of
S in the parametrization X. Since Xij = Xji, we conclude that Γ^{k}_{ij} = Γ^{k}_{ji}; that is, the Christoffel
symbols are symmetric relative to the lower indices.

To determine the Christoffel symbols, we take the inner product of the X_{ij} with X_{k} and use the
definition of (g_{ij}) and (g^{ij}) to obtain

hX_{ij}, X_{k}i = ∂

∂u_{j}hX_{i}, X_{k}i − hX_{i}, X_{kj}i = ∂g_{ik}

∂u_{j} − hX_{i}, X_{kj}i = g_{ik,j}− hX_{i}, X_{kj}i,
where g_{ik,j} = ∂g_{ik}

∂uj

⇐⇒(∗) h

2

X

`=1

Γ^{`}_{ij}X`, Xki = gik,j− hXi, Xkji

⇐⇒

2

X

`=1

Γ^{`}_{ij}g_{`k} = g_{ik,j}− hX_{i}, X_{kj}i

⇐⇒(†) 2

X

k=1 2

X

`=1

Γ^{`}_{ij}g^{mk}g_{`k} =

2

X

k=1

g^{mk}g_{ik,j}−

2

X

k=1

g^{mk}hX_{i}, X_{kj}i, m = 1, 2

⇐⇒(†) 2

X

`=1

δ_{m`}Γ^{`}_{ij} =

2

X

k=1

g^{mk}g_{ik,j} −

2

X

k=1

g^{mk}hX_{i}, X_{kj}i, m = 1, 2

⇐⇒(†) Γ^{m}_{ij} =

2

X

k=1

g^{mk}g_{ik,j}−

2

X

k=1

g^{mk}hX_{i}, X_{kj}i, m = 1, 2

Γ^{m}_{ij}=Γ^{m}_{ji}

⇐⇒ Γ^{m}_{ji} =

2

X

k=1

g^{mk}g_{jk,i}−

2

X

k=1

g^{mk}hX_{j}, X_{ki}i, m = 1, 2

=⇒ 2Γ^{m}_{ij} =

2

X

k=1

g^{mk}(gik,j + gjk,i) −

2

X

k=1

g^{mk} ∂

∂u_{k}hXi, Xji =

2

X

k=1

g^{mk}(gik,j + gjk,i− gij,k), m = 1, 2

=⇒ Γ^{m}_{ij} = 1
2

2

X

k=1

g^{mk}(g_{ik,j}+ g_{jk,i}− g_{ij,k}), m, i, j = 1, 2.

Since X : U ⊂ R^{2} → R^{3} is differentiable,
(X_{ii})_{j} = (X_{ij})_{i}, 1 ≤ i 6= j ≤ 2

⇐⇒

2

X

k=1

Γ^{k}_{ii}X_{k}+ h_{ii}N

!

j

=

2

X

k=1

Γ^{k}_{ij}X_{k}+ h_{ij}N

!

i

⇐⇒

2

X

k=1

(Γ^{k}_{ii})_{j}X_{k}+

2

X

k=1

Γ^{k}_{ii}X_{kj}+ h_{ii,j}N + h_{ii}N_{j} =

2

X

k=1

(Γ^{k}_{ij})_{i}X_{k}+

2

X

k=1

Γ^{k}_{ij}X_{ki}+ h_{ij,i}N + h_{ij}N_{i},

where (Γ^{k}_{ii})_{j} = ∂Γ^{k}_{ii}

∂u_{j}, h_{ij,i} = ∂h_{ij}

∂u_{i}

⇐⇒

2

X

k=1

(Γ^{k}_{ii})_{j}X_{k}+

2

X

k, `=1

Γ^{k}_{ii}Γ^{`}_{kj}X_{`}+

2

X

k=1

Γ^{k}_{ii}h_{kj}N + h_{ii,j}N +

2

X

k=1

h_{ii}a_{kj}X_{k}

=

2

X

k=1

(Γ^{k}_{ij})iXk+

2

X

k, `=1

Γ^{k}_{ij}Γ^{`}_{ki}X`+

2

X

k=1

Γ^{k}_{ij}hkiN + hij,iN +

2

X

k=1

hijakiXk

⇐⇒

2

X

k=1

(Γ^{k}_{ii})_{j}X_{k}+

2

X

`, k=1

Γ^{`}_{ii}Γ^{k}_{`j}X_{k}+

2

X

k=1

Γ^{k}_{ii}h_{kj}N + h_{ii,j}N +

2

X

k=1

h_{ii}a_{kj}X_{k} k ↔ ` in double sum

=

2

X

k=1

(Γ^{k}_{ij})_{i}X_{k}+

2

X

`, k=1

Γ^{`}_{ij}Γ^{k}_{`i}X_{k}+

2

X

k=1

Γ^{k}_{ij}h_{ki}N + h_{ij,i}N +

2

X

k=1

h_{ij}a_{ki}X_{k} k ↔ ` in double sum

=⇒ 0 =

2

X

k=1

"

(Γ^{k}_{ij})_{i}− (Γ^{k}_{ii})_{j} +

2

X

`=1

Γ^{`}_{ij}Γ^{k}_{`i}−

2

X

`=1

Γ^{`}_{ii}Γ^{k}_{`j} + h_{ij}a_{ki}− h_{ii}a_{kj}

#
X_{k}

+ h_{ij,i}− h_{ii,j}+

2

X

k=1

Γ^{k}_{ij}h_{ki}−

2

X

k=1

Γ^{k}_{ii}h_{kj}

! N

⇐⇒ (Γ^{k}_{ij})_{i}− (Γ^{k}_{ii})_{j} +

2

X

`=1

Γ^{`}_{ij}Γ^{k}_{`i}−

2

X

`=1

Γ^{`}_{ii}Γ^{k}_{`j} = h_{ii}a_{kj}− h_{ij}a_{ki} 1 ≤ i 6= j ≤ 2,

and hij,i− hii,j +

2

X

k=1

Γ^{k}_{ij}hki−

2

X

k=1

Γ^{k}_{ii}hkj = 0 1 ≤ i 6= j ≤ 2 called Mainardi-Codazzi equations

Since h_{ij} = h_{ji}, 1 ≤ i 6= j ≤ 2,

h^{11} h^{12}
h^{21} h^{22}

=h_{11} h_{12}
h21 h22

−1

= 1

eg − f^{2}

h_{22} −h_{21}

−h12 h11

= 1

eg − f^{2}

h_{22} −h_{12}

−h21 h11

, we have

h_{ii}a_{kj}− h_{ij}a_{ki} = (eg − f^{2})[h^{jj}a_{kj}+ h^{ji}a_{ki}] = (eg − f^{2})

2

X

`=1

h^{j`}a_{k`}

= (eg − f^{2}) ×h^{11} h^{12}
h^{21} h^{22}

·a_{11} a_{21}
a_{12} a_{22}

jk

(the jk-entry of (h^{mn}) · (a_{pq}))

= eg − f^{2}

EG − F^{2} ×h^{11} h^{12}
h^{21} h^{22}

·h_{11} h_{12}
h_{21} h_{22}

·−g_{22} g_{21}
g_{12} −g_{11}

jk

= K ×−g_{22} g_{21}
g_{12} −g_{11}

jk

= K ×−G F F −E

jk

and the Gauss curvature formulas
(Γ^{k}_{ij})_{i}− (Γ^{k}_{ii})_{j} +

2

X

`=1

Γ^{`}_{ij}Γ^{k}_{`i}−

2

X

`=1

Γ^{`}_{ii}Γ^{k}_{`j} = h_{ii}a_{kj}− h_{ij}a_{ki} = K ×−g_{22} g_{21}
g12 −g11

jk

,

where Γ^{k}_{ij} = 1
2

2

X

`=1

g^{k`}(g_{i`,j}+ g_{j`,i}− g_{ij,`}), i, j, k = 1, 2.

In particular, we obtain the following when
j = k = 2, i = 1 =⇒ (Γ^{2}_{12})_{1}− (Γ^{2}_{11})_{2}+

2

X

`=1

Γ^{`}_{12}Γ^{2}_{`1}−

2

X

`=1

Γ^{`}_{11}Γ^{2}_{`2} = −h_{12}a_{21}+ h_{11}a_{22}= −EK,

j = k = 1, i = 2 =⇒ (Γ^{1}_{21})_{2}− (Γ^{1}_{22})_{1}+

2

X

`=1

Γ^{`}_{21}Γ^{1}_{`2}−

2

X

`=1

Γ^{`}_{22}Γ^{1}_{`1} = h_{22}a_{k1}− h_{21}a_{k2} = −GK,

j 6= k = 1, i = 1 =⇒ (Γ^{1}_{12})_{1}− (Γ^{1}_{11})_{2}+

2

X

`=1

Γ^{`}_{12}Γ^{1}_{`1}−

2

X

`=1

Γ^{`}_{11}Γ^{1}_{`2} = −h_{12}a_{11}+ h_{11}a_{12}= F K.

Example Let X(u_{1}, u_{2}) be an orthogonal parametrization (that is, F = g_{12} = g_{21} = 0) of a
neighborhood of an oriented surface S. Let g_{,m}^{ik} = ∂g^{ik}

∂u_{m} and g_{k`,m} = ∂gk`

∂u_{m}. Since

2

X

k=1

g^{ik}g_{kj} = δ_{ij} =

(1 if i = j,

0 if i 6= j., g_{12} = g_{21}= 0 and g^{12}= g^{21}= 0,

and using Γ^{k}_{ij} = 1
2

2

X

k=1

g^{k`}(g_{j`,i}+ g_{`i,j}− g_{ij,`}) = 1

2g^{kk}(g_{jk,i}+ g_{ki,j}− g_{ij,k}), we have

2

X

k=1

g^{ik}_{,m}g_{k`}+

2

X

k=1

g^{ik}g_{k`,m}= 0 =⇒

2

X

`=1

g^{`j}

2

X

k=1

g_{,m}^{ik}g_{k`}+

2

X

`=1

g^{`j}

2

X

k=1

g^{ik}g_{k`,m}= 0

=⇒ g^{ij}_{,m}=

2

X

k=1

g^{ik,m}δ_{jk} = −

2

X

k, `=1

g^{ik}g_{k`,m}g^{`j} =⇒ g_{,m}^{ij} = −g^{ii}g_{ij,m}g^{jj} =⇒ g^{ii}_{,m}= −g^{ii}g_{ii,m}g^{ii},

=⇒ Γ^{2}_{12} = 1

2g^{22}g_{22,1}, Γ^{2}_{11} = −1

2g^{22}g_{11,2}, Γ^{1}_{12} = 1

2g^{11}g_{11,2}, Γ^{1}_{11}= 1

2g^{11}g_{11,1}, Γ^{2}_{22}= 1

2g^{22}g_{22,2}
and

Γ^{2}_{12}

1− Γ^{2}_{11}

2 +

2

X

`=1

Γ^{`}_{12}Γ^{2}_{`1}−

2

X

`=1

Γ^{`}_{11}Γ^{2}_{`2}= −Kg_{11}

⇐⇒ 1

2 g^{22}g_{22,1}

1+1

2 g^{22}g_{11,2}

2 −1

4 g^{11}g_{11,2}g^{22}g_{11,2} +1

4 g^{22}g_{22,1}g^{22}g_{22,1}

−1

4 g^{11}g_{11,1}g^{22}g_{22,1} +1

4 g^{22}g_{11,2}g^{22}g_{22,2} = −Kg_{11}

⇐⇒

g_{22,1}
2√

g11g22

√g_{11}

√g22

1

+

g_{11,2}
2√

g11g22

√g_{11}

√g22

2

+ (g_{22,1})^{2}+ g_{11,2}g_{22,2}

4 (g22)^{2} − (g_{11,2})^{2}+ g_{11,1}g_{22,1}
4 g11g22

= −Kg_{11}

⇐⇒

g_{22,1}
2√

g_{11}g_{22}

1

√g_{11}

√g_{22} +

g_{11,2}
2√

g_{11}g_{22}

2

√g_{11}

√g_{22} = −Kg_{11}

⇐⇒ K = − 1

2√
g_{11}g_{22}

g_{22,1}

√g_{11}g_{22}

1

+

g_{11,2}

√g_{11}g_{22}

2

Parallel Transport. Geodesics.

Definition Let w : U → R^{3} be a differentiable tangent vector field in an open set U ⊂ S and
p ∈ U. Let y ∈ TpS. Consider a parametrized curve

α : (−ε, ε) → U, with α(0) = p and α^{0}(0) = y,

and let w(t) = w(α(t)) ∈ T_{α(t)}S, t ∈ (−ε, ε), be the restriction of the vector field w to the curve
α.

Then the covariant derivative at p of the vector field w relative to the vector y, denoted Dw
dt (0)
or D_{y}w(p), is defined to be the normal projection of dw

dt(0) onto the plane T_{p}S, i.e.

Dw

dt (0) = dw

dt (0) − hdw

dt (0), N (p)iN.

In terms of a parametrization X(u_{1}, u_{2}) of U ⊂ S at p, let X(u_{1}(t), u_{2}(t)) = α(t) ⊂ S and

w(t) =a_{1}(u_{1}(t), u_{2}(t))X_{u}_{1} + a_{2}(u_{1}(t), u_{2}(t))X_{u}_{2} = a_{1}(t)X_{1}+ a_{2}(t)X_{2} =

2

X

i=1

a_{i}X_{i} ∈ T_{α(t)}S

be the expression of α(t) and w(t) in the parametrization X(u, v), respectively. Then dw

dt =

2

X

i, j=1

a_{i}X_{ij}u^{0}_{j}+

2

X

i=1

a^{0}_{i}X_{i} =

2

X

i, j,k=1

a_{i}u^{0}_{j}Γ^{k}_{ij}X_{k}+

2

X

i, j=1

a_{i}u^{0}_{j}h_{ij}N +

2

X

k=1

a^{0}_{k}X_{k}

and the covariant derivative of w at t is given by Dw

dt =

2

X

i, j,k=1

a_{i}u^{0}_{j}Γ^{k}_{ij}X_{k}+

2

X

i=1

a^{0}_{i}X_{i}

=

2

X

k=1

a^{0}_{k}+

2

X

i, j=1

Γ^{k}_{ij}a_{i}u^{0}_{j}

!

X_{k} ∈ T_{α(t)}S

Note that the covariant differentiation Dw

dt depends only on the vector (u^{0}_{1}, u^{0}_{2}), the coordinates of
α^{0}(t) in the basis {X_{1}, X_{2}}, and not on the curve α. Also since it depends only on the Christoffel
symbols, that is, the first fundamental form of the surface, the covariant differentiation Dw

dt is a concept of intrinsic geometry.

Definition A vector field w ∈ T_{α}S along a parametrized curve α : I → S is said to be parallel
if Dw

dt = 0 for every t ∈ I.

Example In a plane P, since Γ^{k}_{ij} = 0, 1 ≤ i, j, k ≤ 2, the notion of parallel field w = a_{1}X_{1}+ a_{2}X_{2}
along a parametrized curve α ⊂ P reduces to that of a constant field, i.e. a^{0}_{1} = a^{0}_{2} = 0, along α;

that is, the length of the vector and its angle with a fixed direction are constant.

Those properties are partially reobtained on any surface as the following proposition shows.

Proposition Let w, v ∈ T_{α}S be parallel vector fields along α : I → S. Then hw(t), v(t)i is
constant for all t ∈ I. In particular, the lengths |w(t)| and |v(t)| are constant, and the angle

∠(v(t), w(t)) between w(t), v(t) ∈ Tα(t)S is constant for all t ∈ I.

Proof Since w(t), v(t) ∈ T_{α(t)}S and Dw

dt = Dv

dt = 0, we have d

dthw(t), v(t)i = hdw

dt , v(t)i + hw(t),dv

dti = hDw

dt , v(t)i + hw(t), Dv dt i = 0,

and hw(t), v(t)i = constant for all t ∈ I and for any parallel vector fields w and v along α.

Proposition Let α : I → S be a parametrized curve in S and let w_{0} ∈ T_{α(t}_{0}_{)}S, t_{0} ∈ I. Then
there exists a unique parallel vector field w(t) = a_{1}(t)X_{1}(u_{1}(t), u_{2}(t))+a_{2}(t)X_{2}(u_{1}(t), u_{2}(t)) along
α(t), with w(t_{0}) = w_{0}, i.e. there is a unique solution to the initial-value problem

a^{0}_{k}+

2

X

i, j=1

Γ^{k}_{ij}a_{i}u^{0}_{j} = 0, k = 1, 2, with a_{1}X_{1}+ a_{2}X_{2}|_{t=t}_{0} = w(t_{0}) = w_{0}.

Definition Let α : I → S be a parametrized curve and w_{0} ∈ T_{α(t}_{0}_{)}S, t_{0} ∈ I. Let w be a paralle
vector field along α, with w(t_{0}) = w_{0}. The vector w(t_{1}), t_{1} ∈ I, is called the parallel transport of
w0 along α at the point t1.

Definition A nonconstant, parametrized curve γ : I → S is said to be geodesic at t ∈ I if the
field of its tangent vectors γ^{0}(t) is parallel along γ at t; that is

Dγ^{0}(t)
dt = 0;

γ is a parametrized geodesic if it is geodesic for all t ∈ I, i.e. γ(t) = X(u_{1}(t), u_{2}(t)), t ∈ I is a
geodesic if γ^{0}(t) = u^{0}_{1}X_{1}+ u^{0}_{2}X_{2} satisfies the geodesic equations

Dγ^{0}(t)

dt = 0 ⇐⇒ u^{00}_{k}+

2

X

i,j=1

Γ^{k}_{ij}u^{0}_{i}u^{0}_{j}, k = 1, 2. (∗)

Examples

(1) If S is a plane, then S can be parametrized by X(u_{1}, u_{2}) with X_{ij} = X_{u}_{i}_{u}_{j} = 0 ∈ R^{3}
everywhere in S, 1 ≤ i, j ≤ 2. This implies that X_{1} = X_{u}_{1} and X_{2} = X_{u}_{2} are constant
vector in S, Γ^{k}_{ij} = 0 for all 1 ≤ i, j, k ≤ 2, and γ(t) = X(u_{1}(t), u_{2}(t)) is a geodesic in a plane
S if

u^{00}_{k}(t) = 0, ∀ t ∈ I =⇒ u^{0}_{k}(t) = c_{k} (a constant) ∀ t ∈ I =⇒ u_{k}(t) = c_{k}t+d_{k} ∀ t ∈ I, k = 1, 2
Hence γ is a geodesic in a plane S if and only if γ is a straight line in S.

(2) Let γ(u_{2}) = (f (u_{2}), 0, g(u_{2})), f (u_{2}) 6= 0, a < u_{2} < b, be a regular curve and S be a surface
of revolution with the parametrization

X(u_{1}, u_{2}) = (f (u_{2}) cos u_{1}, f (u_{2}) sin u_{1}, g(u_{2})), 0 < u_{1} < 2π, a < u_{2} < b.

Then the matrix (g_{ij}) and its inverse (g^{ij}) of the first fundamental form

2

X

i,j=1

g_{ij}u^{0}_{i}u^{0}_{j} are
given by

g_{11} g_{12}
g_{21} g_{22}

=f^{2} 0

0 (f^{0})^{2}+ (g^{0})^{2}

⇐⇒ g^{11} g^{12}
g^{21} g^{22}

=f^{−2} 0

0 (f^{0})^{2}+ (g^{0})^{2}^{−1}

where f and g are functions of u_{2}and the Christoffel symbols Γ^{k}_{ij} = 1
2

2

X

`=1

g^{k`}(g_{j`,i}+g_{`i,j}−g_{ij,`})
are given by

Γ^{1}_{ij} = 1

2f^{−2}(g_{j1,i}+ g_{1i,j} − g_{ij,1}) and Γ^{2}_{ij} = 1

2(f^{0})^{2}+ (g^{0})^{2}−1

(g_{j2,i}+ g_{2i,j} − g_{ij,2})
and

Γ^{1}_{11} Γ^{1}_{12}
Γ^{1}_{21} Γ^{1}_{22}

=

0 f f^{0}
f^{2}
f f^{0}

f^{2} 0

and Γ^{2}_{11} Γ^{2}_{12}
Γ^{2}_{21} Γ^{2}_{22}

=

− f f^{0}

(f^{0})^{2}+ (g^{0})^{2} 0
0 f^{0}f^{00}+ g^{0}g^{00}

(f^{0})^{2}+ (g^{0})^{2}

this implies that X(u_{1}(t), u_{2}(t)) is a geodesic of the surface of revolution S if u_{1}, u_{2} satisfy
the system ofequations

u^{00}_{1} +2f f^{0}

f^{2} u^{0}_{1}u^{0}_{2} = 0 and u^{00}_{2} − f f^{0}

(f^{0})^{2}+ (g^{0})^{2}(u^{0}_{1})^{2}+ f^{0}f^{00}+ g^{0}g^{00}

(f^{0})^{2}+ (g^{0})^{2}(u^{0}_{2})^{2} = 0, (††)

where u^{0}_{k} = du_{k}

dt , f^{0} = df

du_{2} and g^{0} = dg
du_{2}.

If the meridian γ(s) = {X(u_{1}, u_{2}) | u_{1} = constant, u_{2} = u_{2}(s)} is parametrized by arc
length s, then the 1^{st} equation of (††)holds, and, since γ^{0}(s) = X_{1}u^{0}_{1}+ X_{2}u^{0}_{2} = X_{2}u^{0}_{2},
1 = hγ^{0}(s), γ^{0}(s)i = I_{p}(γ^{0}(s)) = hX_{1}u^{0}_{1}+X_{2}u^{0}_{2}, X_{1}u^{0}_{1}+X_{2}u^{0}_{2}i = g_{22}(u^{0}_{2})^{2} =(f^{0})^{2}+(g^{0})^{2}(u^{0}_{2})^{2},
we have

(u^{0}_{2})^{2} ^{(∗)}= 1

(f^{0})^{2}+ (g^{0})^{2} =⇒ u^{0}_{2}u^{00}_{2} = − f^{0}f^{00}+ g^{0}g^{00}

(f^{0})^{2}+ (g^{0})^{2}2 u^{0}_{2}

by differentiating both sides with respect to s and using the Chain Rule to get d

dsf^{0} = f^{00}u^{0}_{2}
and d

dsg^{0} = g^{00}u^{0}_{2}. Multiplying both sides by u^{0}_{2}, we get
(u^{0}_{2})^{2}u^{00}_{2} = − f^{0}f^{00}+ g^{0}g^{00}

(f^{0})^{2}+ (g^{0})^{2}2 (u^{0}_{2})^{2} =⇒^{(∗)} u^{00}_{2} = −f^{0}f^{00}+ g^{0}g^{00}

(f^{0})^{2}+ (g^{0})^{2}(u^{0}_{2})^{2} the 2^{nd} equation of (††)
and this implies that arc length parametrized meridians are geodesics.

If the parallel γ(s) = {X(u_{1}, u_{2}) | u_{2} = constant, u_{1} = u_{1}(s)} is parametrized by arc
length s, since 1 = Ip(γ^{0}(s)) = (f (u2))^{2}(u^{0}_{1})^{2}, we have (u^{0}_{1})^{2} = 1/f (u2) = constant 6= 0
which implies that 2u^{0}_{1}u^{00}_{1} = 0 =⇒ u^{00}_{1} = 0, i.e. the 1^{st} equation of (††) holds, so the arc
length parametrized parallels are geodesics if it satisfies the 2^{nd} equation of (††)

f f^{0}

(f^{0})^{2}+ (g^{0})^{2}(u^{0}_{1})^{2} = 0 =⇒ f^{0} = 0 since f 6= 0, u^{0}_{1} 6= 0.

Definition Let w be a differentiable field of unit vectors along a parametrized curve α : I → S on an oriented surface S. Sincew(t), t ∈ I, is a unit vector field,

dw

dt (t) ⊥ w(t) =⇒ Dw

dt = Dw dt

(N ∧ w(t)),

where the real number Dw dt

is called the algebraic value of the covariant derivative of w at t.

Definition Let C be an oriented regular curve contained in an oriented surface S, and let α(s)
be a parametrization of C, in a neighborhood of p ∈ S, by the arc length s. The algebraic value
of the covariant derivative of α^{0}(s) at p, Dα^{0}(s)

ds

= k_{g} is called the geodesic curvature of C at
p.

Remark The geodesics which are regular curves are thus characterized as curves whose geodesic curvature is zero and note that the geodesic curvature of C ⊂ S changes sign when we change the orientation of either C or S.