the diffeomorphism ϕ : S → ¯S prserves the first fundamental form

Isometries; Conformal Maps

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

hw₁, w₂i_p = hdϕ_p(w₁), dϕ_p(w₂)i_ϕ(p) ∀ p ∈ S, ∀ w₁, w₂ ∈ T_pS.

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_pS → 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_pS, 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 2hw₁, w₂i = I_p(w₁+ w₂) − I_p(w₁) − I_p(w₂) ∀ w₁, w₂ ∈ T_pS

= I_ϕ(p)(dϕ_p(w₁+ w₂)) − I_ϕ(p)(dϕ_p(w₁)) − I_ϕ(p)(dϕ_p(w₂))

= hdϕ(w₁), dϕ(w₂)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) ∈ R³ | z = 0} and the cylinder S = {(x, y, z) ∈ R³ | x²+ y² = 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 C⁰, e.g. C⁰ = {(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) | x²+ y² = 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_pS. 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_uu⁰+ X_vv⁰.

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_uu⁰ + ¯X_vv⁰. Since

Ip(w) = E(u⁰)²+ 2F u⁰v⁰+ G(v⁰)², I_ϕ(p)(dϕ_p(w)) = E(u¯ ⁰)²+ 2 ¯F u⁰v⁰+ ¯G(v⁰)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(w₁), dϕ_p(w₂)i = λ²(p) hw₁, w₂i^w=⇒ |dϕ^1=w2 _p(w₁)|² = λ²(p) |w₁|² ∀ p ∈ S, ∀ w₁, w₂ ∈ T_pS, where λ² 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 = hα⁰, β⁰i

|α⁰| |β⁰|, 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(α⁰), dϕ(β⁰)i

|dϕ(α⁰)| |dϕ(β⁰)| = λ²hα⁰, β⁰i

λ²|α⁰| |β⁰| = cos θ.

Proposition Suppose that there exist parametrizations X : U → S and ¯X : U → ¯S such that E = λ²E, F = λ¯ ²F , G = λ¯ ²G 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 = a²cosh²v, F = 0, G = a²(1 + sinh²v) = a²cosh²v ∀ (u, v) ∈ U, E = a¯ ²+ ¯v², 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 = a²cosh²v, F = 0, G = a²cosh²v ∀ (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

x²+ y², k > 0, (x, y) 6= (0, 0), and let U ⊂ R² 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 ⊂ R³ 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

x²+ y² = cot α q

ρ²sin²α = 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_ρi = 1, F = hF_ρ, F_θi = 0, G = hF_θ, F_θi = ρ², 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_ρi = 1 = E, F = h ¯¯ X_ρ, ¯X_θi = 0 = F, G = h ¯¯ X_θ, ¯X_θi = ρ² = 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 = λ²(u, v), F = 0, G = λ²(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 ⊂ R² → S be a parametrization in the orientation of S. At each p ∈ X(U ), since X_u, X_v, N ∈ R³ are linearly independent, we may express vectors X_uu, X_uv, X_vu, X_vv, N_u, N_v ∈ R³ in the basis {Xu, Xv, N } and obtain

X_uu = Γ¹₁₁X_u+ Γ²₁₁X_v+ eN X_uv = Γ¹₁₂X_u+ Γ²₁₂X_v+ f N X_vu = Γ¹₂₁X_u+ Γ²₂₁X_v+ f N Xvv = Γ¹₂₂Xu+ Γ²₂₂Xv+ gN

N_u = a₁₁X_u + a₂₁X_v N_v = a₁₂X_u + a₂₂X_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 Γ¹₁₂ = Γ¹₂₁ and Γ²₁₂ = Γ²₂₁; 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







 Γ¹₁₁ Γ²₁₁



=





hX_uu, X_ui hX_uu, X_vi



=









 1 2Eu

F_u− 1 2E_v











=⇒



 Γ¹₁₁ Γ²₁₁



=



 E F

F G





−1









 1 2Eu

F_u− 1 2E_v













 Γ¹₁₂ Γ²₁₂



=





hX_uv, X_ui hX_uv, X_vi



=









 1 2E_v 1 2G_u











=⇒



 Γ¹₂₁ Γ²₂₁



=



 Γ¹₁₂ Γ²₁₂



=



 E F

F G





−1









 1 2E_v 1 2G_u













 E F

F G







 Γ¹₂₂ Γ²₂₂



=





hX_vv, X_ui hX_vv, X_vi



=











F_v− 1 2G_u 1 2G_v











=⇒



 Γ¹₂₂ Γ²₂₂



=



 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_vi = 0 at each p ∈ X(U ), then

Γ¹₁₁ Γ²₁₁



= 1

2(EG − F²)

 GE_u

−EE_v



Γ¹₁₂ Γ²₁₂



= 1

2(EG − F²)

 GE_v EG_u



Γ¹₂₂ Γ²₂₂



= 1

2(EG − F²)

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 = (f⁰(v))²+ (g⁰(v))², we obtain

E_u = 0, E_v = 2f f⁰, F_u = F_v = 0, G_u = 0, G_v = 2(f⁰f⁰⁰+ g⁰g⁰⁰), and

Γ¹₁₁= 0, Γ²₁₁ = − f f⁰

(f⁰)² + (g⁰)², Γ¹₁₂= f f⁰

f² , Γ²₁₂= 0, Γ¹₂₂ = 0, Γ²₂₂= f⁰f⁰⁰+ g⁰g⁰⁰ (f⁰)²+ (g⁰)². Since X : U ⊂ R² → R³ is differentiable,

(X_uu)_v = (X_uv)_u,

⇐⇒ Γ¹₁₁X_u+ Γ²₁₁X_v+ eN

v = Γ¹₁₂X_u+ Γ²₁₂X_v+ f N

u

⇐⇒ Γ¹₁₁X_uv+ Γ²₁₁X_vv+ eN_v+ (Γ¹₁₁)_vX_u+ (Γ²₁₁)_vX_v+ e_vN

=Γ¹₁₂X_uu+ Γ²₁₂X_vu+ f N_u+ (Γ¹₁₂)_uX_u+ (Γ²₁₂)_uX_v + f_uN (∗) By equating the coefficients of X_v, and using

a₁₁ a₂₁ a12 a22



= − e f f g

 E F

F G

−1

= −1

EG − F²

 eG − f F −eF + f E f G − gF −f F + gE

 . we obtain the following formula for the Gaussian curvature K

Γ¹₁₁Γ²₁₂+ Γ²₁₁Γ²₂₂+ ea22+ (Γ²₁₁)v = Γ¹₁₂Γ²₁₁+ Γ²₁₂Γ²₁₂+ f a21+ (Γ²₁₂)u

⇐⇒ (Γ²₁₂)_u− (Γ²₁₁)_v+ Γ¹₁₂Γ²₁₁+ Γ²₁₂Γ²₁₂− Γ¹₁₁Γ²₁₂− Γ²₁₁Γ²₂₂= ea₂₂− f a₂₁

⇐⇒ (Γ²₁₂)u− (Γ²₁₁)v+ Γ¹₁₂Γ²₁₁+ Γ²₁₂Γ²₁₂− Γ¹₁₁Γ²₁₂− Γ²₁₁Γ²₂₂= −E eg − f² EG − F²

⇐⇒ (Γ²₁₂)_u− (Γ²₁₁)_v+ Γ¹₁₂Γ²₁₁+ Γ²₁₂Γ²₁₂− Γ¹₁₁Γ²₁₂− Γ²₁₁Γ²₂₂= −EK,

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

Remarks

• By equating the coefficients of X_uin equation(∗), we obtain another formula of the Gaussian curvature K.

(Γ¹₁₂)u− (Γ¹₁₁)v+ Γ¹₁₂Γ¹₁₁+ Γ²₁₂Γ¹₁₂− Γ¹₁₁Γ¹₁₂− Γ²₁₁Γ¹₂₂ = (Γ¹₁₂)u− (Γ¹₁₁)v + Γ²₁₂Γ¹₁₂− Γ²₁₁Γ¹₂₂= F K.

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

e_v − f_u = eΓ¹₁₂+ f (Γ²₁₂− Γ¹₁₁) − gΓ²₁₁ (†).

• By equating the coefficients of N in equation (X_vv)_u− (X_vu)_v = 0, we obtain f_v− g_u = eΓ¹₂₂+ f (Γ²₂₂− Γ¹₁₂) − gΓ²₁₂ (††).

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 ⊂ R², with E > 0 and G > 0. Assume that

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

• and that EG − F² > 0.

Then,

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

• and a diffeomorphism X : U → X(U ) ⊂ R³

such that the regular surface X(U ) ⊂ R³ 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 ) ⊂ R³

is another diffeomorphism satisfying the same conditions, then there exist a translation T and a proper linear orthogonal transformation ρ in R³ 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 u₁u₂-plane, and X : U ⊂ R² → S be a parametrization in the orientation of S. At each p ∈ X(U ), let X1 = Xu1, X2 = Xu2, and let

g₁₁ = hX₁, X₁i = E, g₁₂ = g₂₁= hX₁, X₂i = F, g₂₂ = hX₂, X₂i = G ⇐⇒ g₁₁ g₁₂ g₂₁ g₂₂



=E F

F G

 , and

g¹¹ g¹² g²¹ g²²



= (g^ij) = (g_ij)⁻¹ = 1 det(g_ij)

 g₂₂ −g₁₂

−g₂₁ g₁₁



= 1

EG − F²

 G −F

−F E

 . Note that

2

X

k=1

g^mkg_{k`</sub> <sup>(†)</sup>= δ<sub>m`} =

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

Since X1, X2, N ∈ R³ are linearly independent, we may express vectors Xij = Xuiuj ∈ R³ and N_i = N_ui ∈ T_pS as

X_ij ^(∗)=

2

X

k=1

Γ^k_ijX_k+ h_ijN, i, j = 1, 2, where h₁₁ h₁₂ h₂₁ h₂₂



= e f f g



Ni (∗∗)=

2

X

j=1

ajiXj, i = 1, 2,

where

a₁₁ a₂₁ a₁₂ a₂₂



= −h₁₁ h₁₂ h₂₁ h₂₂

 g¹¹ g¹² g²¹ g²²



= 1

EG − F²

h₁₁ h₁₂ h₂₁ h₂₂

 −g₂₂ g₂₁ g₁₂ −g₁₁

 ,

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_ki = ∂

∂u_jhX_i, X_ki − hX_i, X_kji = ∂g_ik

∂u_j − hX_i, X_kji = g_ik,j− hX_i, X_kji, where g_ik,j = ∂g_ik

∂uj

⇐⇒(∗) h

2

X

`=1

Γ<sup>`</sup><sub>ij</sub>X`, Xki = gik,j− hXi, Xkji

⇐⇒

2

X

`=1

Γ<sup>`</sup><sub>ij</sub>g<sub>`k</sub> = g_ik,j− hX_i, X_kji

⇐⇒(†) 2

X

k=1 2

X

`=1

Γ<sup>`</sup><sub>ij</sub>g<sup>mk</sup>g<sub>`k</sub> =

2

X

k=1

g^mkg_ik,j−

2

X

k=1

g^mkhX_i, X_kji, m = 1, 2

⇐⇒(†) 2

X

`=1

δ<sub>m`</sub>Γ<sup>`</sup>_ij =

2

X

k=1

g^mkg_ik,j −

2

X

k=1

g^mkhX_i, X_kji, m = 1, 2

⇐⇒(†) Γ^m_ij =

2

X

k=1

g^mkg_ik,j−

2

X

k=1

g^mkhX_i, X_kji, m = 1, 2

Γ^m_ij=Γ^m_ji

⇐⇒ Γ^m_ji =

2

X

k=1

g^mkg_jk,i−

2

X

k=1

g^mkhX_j, X_kii, m = 1, 2

=⇒ 2Γ^m_ij =

2

X

k=1

g^mk(gik,j + gjk,i) −

2

X

k=1

g^mk ∂

∂u_khXi, 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 ⊂ R² → R³ is differentiable, (X_ii)_j = (X_ij)_i, 1 ≤ i 6= j ≤ 2

⇐⇒

2

X

k=1

Γ^k_iiX_k+ h_iiN

!

j

=

2

X

k=1

Γ^k_ijX_k+ h_ijN

!

i

⇐⇒

2

X

k=1

(Γ^k_ii)_jX_k+

2

X

k=1

Γ^k_iiX_kj+ h_ii,jN + h_iiN_j =

2

X

k=1

(Γ^k_ij)_iX_k+

2

X

k=1

Γ^k_ijX_ki+ h_ij,iN + h_ijN_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)_jX_k+

2

X

k, `=1

Γ^k_iiΓ<sup>`</sup><sub>kj</sub>X<sub>`</sub>+

2

X

k=1

Γ^k_iih_kjN + h_ii,jN +

2

X

k=1

h_iia_kjX_k

=

2

X

k=1

(Γ^k_ij)iXk+

2

X

k, `=1

Γ^k_ijΓ<sup>`</sup><sub>ki</sub>X`+

2

X

k=1

Γ^k_ijhkiN + hij,iN +

2

X

k=1

hijakiXk

⇐⇒

2

X

k=1

(Γ^k_ii)_jX_k+

2

X

`, k=1

Γ<sup>`</sup><sub>ii</sub>Γ<sup>k</sup><sub>`j</sub>X_k+

2

X

k=1

Γ^k_iih_kjN + h_ii,jN +

2

X

k=1

h_iia_kjX_k k ↔ ` in double sum

=

2

X

k=1

(Γ^k_ij)_iX_k+

2

X

`, k=1

Γ<sup>`</sup><sub>ij</sub>Γ<sup>k</sup><sub>`i</sub>X_k+

2

X

k=1

Γ^k_ijh_kiN + h_ij,iN +

2

X

k=1

h_ija_kiX_k k ↔ ` in double sum

=⇒ 0 =

2

X

k=1

"

(Γ^k_ij)_i− (Γ^k_ii)_j +

2

X

`=1

Γ<sup>`</sup><sub>ij</sub>Γ<sup>k</sup><sub>`i</sub>−

2

X

`=1

Γ<sup>`</sup><sub>ii</sub>Γ<sup>k</sup><sub>`j</sub> + h_ija_ki− h_iia_kj

# X_k

+ h_ij,i− h_ii,j+

2

X

k=1

Γ^k_ijh_ki−

2

X

k=1

Γ^k_iih_kj

! N

⇐⇒ (Γ^k_ij)_i− (Γ^k_ii)_j +

2

X

`=1

Γ<sup>`</sup><sub>ij</sub>Γ<sup>k</sup><sub>`i</sub>−

2

X

`=1

Γ<sup>`</sup><sub>ii</sub>Γ<sup>k</sup><sub>`j</sub> = h_iia_kj− h_ija_ki 1 ≤ i 6= j ≤ 2,

and hij,i− hii,j +

2

X

k=1

Γ^k_ijhki−

2

X

k=1

Γ^k_iihkj = 0 1 ≤ i 6= j ≤ 2 called Mainardi-Codazzi equations

Since h_ij = h_ji, 1 ≤ i 6= j ≤ 2,

h¹¹ h¹² h²¹ h²²



=h₁₁ h₁₂ h21 h22

−1

= 1

eg − f²

 h₂₂ −h₂₁

−h12 h11



= 1

eg − f²

 h₂₂ −h₁₂

−h21 h11

 , we have

h_iia_kj− h_ija_ki = (eg − f²)[h^jja_kj+ h^jia_ki] = (eg − f²)

2

X

`=1

h<sup>j`</sup>a<sub>k`</sub>

= (eg − f²) ×h¹¹ h¹² h²¹ h²²



·a₁₁ a₂₁ a₁₂ a₂₂



jk

(the jk-entry of (h^mn) · (a_pq))

= eg − f²

EG − F² ×h¹¹ h¹² h²¹ h²²



·h₁₁ h₁₂ h₂₁ h₂₂



·−g₂₂ g₂₁ g₁₂ −g₁₁



jk

= K ×−g₂₂ g₂₁ g₁₂ −g₁₁



jk

= K ×−G F F −E



jk

and the Gauss curvature formulas (Γ^k_ij)_i− (Γ^k_ii)_j +

2

X

`=1

Γ<sup>`</sup><sub>ij</sub>Γ<sup>k</sup><sub>`i</sub>−

2

X

`=1

Γ<sup>`</sup><sub>ii</sub>Γ<sup>k</sup><sub>`j</sub> = h_iia_kj− h_ija_ki = K ×−g₂₂ g₂₁ g12 −g11



jk

,

where Γ^k_ij = 1 2

2

X

`=1

g<sup>k`</sup>(g<sub>i`,j</sub>+ g_{j`,i</sub>− g<sub>ij,`}), i, j, k = 1, 2.

In particular, we obtain the following when j = k = 2, i = 1 =⇒ (Γ²₁₂)₁− (Γ²₁₁)₂+

2

X

`=1

Γ<sup>`</sup><sub>12</sub>Γ<sup>2</sup><sub>`1</sub>−

2

X

`=1

Γ<sup>`</sup><sub>11</sub>Γ<sup>2</sup><sub>`2</sub> = −h₁₂a₂₁+ h₁₁a₂₂= −EK,

j = k = 1, i = 2 =⇒ (Γ¹₂₁)₂− (Γ¹₂₂)₁+

2

X

`=1

Γ<sup>`</sup><sub>21</sub>Γ<sup>1</sup><sub>`2</sub>−

2

X

`=1

Γ<sup>`</sup><sub>22</sub>Γ<sup>1</sup><sub>`1</sub> = h₂₂a_k1− h₂₁a_k2 = −GK,

j 6= k = 1, i = 1 =⇒ (Γ¹₁₂)₁− (Γ¹₁₁)₂+

2

X

`=1

Γ<sup>`</sup><sub>12</sub>Γ<sup>1</sup><sub>`1</sub>−

2

X

`=1

Γ<sup>`</sup><sub>11</sub>Γ<sup>1</sup><sub>`2</sub> = −h₁₂a₁₁+ h₁₁a₁₂= F K.

Example Let X(u₁, u₂) be an orthogonal parametrization (that is, F = g₁₂ = g₂₁ = 0) of a neighborhood of an oriented surface S. Let g_,m^ik = ∂g^ik

∂u_m and g<sub>k`,m</sub> = ∂gk`

∂u_m. Since

2

X

k=1

g^ikg_kj = δ_ij =

(1 if i = j,

0 if i 6= j., g₁₂ = g₂₁= 0 and g¹²= g²¹= 0,

and using Γ^k_ij = 1 2

2

X

k=1

g<sup>k`</sup>(g<sub>j`,i</sub>+ g_{`i,j</sub>− g<sub>ij,`}) = 1

2g^kk(g_jk,i+ g_ki,j− g_ij,k), we have

2

X

k=1

g^ik_,mg_k`+

2

X

k=1

g^ikg_k`,m= 0 =⇒

2

X

`=1

g^`j

2

X

k=1

g_,m^ikg_k`+

2

X

`=1

g^`j

2

X

k=1

g^ikg_k`,m= 0

=⇒ g^ij_,m=

2

X

k=1

g^ik,mδ_jk = −

2

X

k, `=1

g^ikg<sub>k`,m</sub>g<sup>`j</sup> =⇒ g_,m^ij = −gⁱⁱg_ij,mg^jj =⇒ gⁱⁱ_,m= −gⁱⁱg_ii,mgⁱⁱ,

=⇒ Γ²₁₂ = 1

2g²²g_22,1, Γ²₁₁ = −1

2g²²g_11,2, Γ¹₁₂ = 1

2g¹¹g_11,2, Γ¹₁₁= 1

2g¹¹g_11,1, Γ²₂₂= 1

2g²²g_22,2 and

Γ²₁₂

1− Γ²₁₁

2 +

2

X

`=1

Γ<sup>`</sup><sub>12</sub>Γ<sup>2</sup><sub>`1</sub>−

2

X

`=1

Γ<sup>`</sup><sub>11</sub>Γ<sup>2</sup><sub>`2</sub>= −Kg₁₁

⇐⇒ 1

2 g²²g_22,1

1+1

2 g²²g_11,2

2 −1

4 g¹¹g_11,2g²²g_11,2
+1

4 g²²g_22,1g²²g_22,1

−1

4 g¹¹g_11,1g²²g_22,1
+1

4 g²²g_11,2g²²g_22,2
= −Kg₁₁

⇐⇒

 g_22,1 2√

g11g22

√g₁₁

√g22



1

+

 g_11,2 2√

g11g22

√g₁₁

√g22



2

+ (g_22,1)²+ g_11,2g_22,2

4 (g22)² − (g_11,2)²+ g_11,1g_22,1 4 g11g22

= −Kg₁₁

⇐⇒

 g_22,1 2√

g₁₁g₂₂



1

√g₁₁

√g₂₂ +

 g_11,2 2√

g₁₁g₂₂



2

√g₁₁

√g₂₂ = −Kg₁₁

⇐⇒ K = − 1

2√ g₁₁g₂₂

 g_22,1

√g₁₁g₂₂



1

+

 g_11,2

√g₁₁g₂₂



2



Parallel Transport. Geodesics.

Definition Let w : U → R³ 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 α⁰(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_yw(p), is defined to be the normal projection of dw

dt(0) onto the plane T_pS, i.e.

Dw

dt (0) = dw

dt (0) − hdw

dt (0), N (p)iN.

In terms of a parametrization X(u₁, u₂) of U ⊂ S at p, let X(u₁(t), u₂(t)) = α(t) ⊂ S and

w(t) =a₁(u₁(t), u₂(t))X_u1 + a₂(u₁(t), u₂(t))X_u2 = a₁(t)X₁+ a₂(t)X₂ =

2

X

i=1

a_iX_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_iX_iju⁰_j+

2

X

i=1

a⁰_iX_i =

2

X

i, j,k=1

a_iu⁰_jΓ^k_ijX_k+

2

X

i, j=1

a_iu⁰_jh_ijN +

2

X

k=1

a⁰_kX_k

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

dt =

2

X

i, j,k=1

a_iu⁰_jΓ^k_ijX_k+

2

X

i=1

a⁰_iX_i

=

2

X

k=1

a⁰_k+

2

X

i, j=1

Γ^k_ija_iu⁰_j

!

X_k ∈ T_α(t)S

Note that the covariant differentiation Dw

dt depends only on the vector (u⁰₁, u⁰₂), the coordinates of α⁰(t) in the basis {X₁, X₂}, 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_αS 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 = a₁X₁+ a₂X₂ along a parametrized curve α ⊂ P reduces to that of a constant field, i.e. a⁰₁ = a⁰₂ = 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_αS 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 w₀ ∈ T_α(t0)S, t₀ ∈ I. Then there exists a unique parallel vector field w(t) = a₁(t)X₁(u₁(t), u₂(t))+a₂(t)X₂(u₁(t), u₂(t)) along α(t), with w(t₀) = w₀, i.e. there is a unique solution to the initial-value problem

a⁰_k+

2

X

i, j=1

Γ^k_ija_iu⁰_j = 0, k = 1, 2, with a₁X₁+ a₂X₂|_t=t0 = w(t₀) = w₀.

Definition Let α : I → S be a parametrized curve and w₀ ∈ T_α(t0)S, t₀ ∈ I. Let w be a paralle vector field along α, with w(t₀) = w₀. The vector w(t₁), t₁ ∈ 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 γ⁰(t) is parallel along γ at t; that is

Dγ⁰(t) dt = 0;

γ is a parametrized geodesic if it is geodesic for all t ∈ I, i.e. γ(t) = X(u₁(t), u₂(t)), t ∈ I is a geodesic if γ⁰(t) = u⁰₁X₁+ u⁰₂X₂ satisfies the geodesic equations

Dγ⁰(t)

dt = 0 ⇐⇒ u⁰⁰_k+

2

X

i,j=1

Γ^k_iju⁰_iu⁰_j, k = 1, 2. (∗)

Examples

(1) If S is a plane, then S can be parametrized by X(u₁, u₂) with X_ij = X_uiuj = 0 ∈ R³ everywhere in S, 1 ≤ i, j ≤ 2. This implies that X₁ = X_u1 and X₂ = X_u2 are constant vector in S, Γ^k_ij = 0 for all 1 ≤ i, j, k ≤ 2, and γ(t) = X(u₁(t), u₂(t)) is a geodesic in a plane S if

u⁰⁰_k(t) = 0, ∀ t ∈ I =⇒ u⁰_k(t) = c_k (a constant) ∀ t ∈ I =⇒ u_k(t) = c_kt+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(u₂) = (f (u₂), 0, g(u₂)), f (u₂) 6= 0, a < u₂ < b, be a regular curve and S be a surface of revolution with the parametrization

X(u₁, u₂) = (f (u₂) cos u₁, f (u₂) sin u₁, g(u₂)), 0 < u₁ < 2π, a < u₂ < b.

Then the matrix (g_ij) and its inverse (g^ij) of the first fundamental form

2

X

i,j=1

g_iju⁰_iu⁰_j are given by

g₁₁ g₁₂ g₂₁ g₂₂



=f² 0

0 (f⁰)²+ (g⁰)²



⇐⇒ g¹¹ g¹² g²¹ g²²



=f⁻² 0

0 (f⁰)²+ (g⁰)²⁻¹



where f and g are functions of u₂and the Christoffel symbols Γ^k_ij = 1 2

2

X

`=1

g<sup>k`</sup>(g<sub>j`,i</sub>+g_{`i,j</sub>−g<sub>ij,`}) are given by

Γ¹_ij = 1

2f⁻²(g_j1,i+ g_1i,j − g_ij,1) and Γ²_ij = 1

2(f⁰)²+ (g⁰)²−1

(g_j2,i+ g_2i,j − g_ij,2) and

Γ¹₁₁ Γ¹₁₂ Γ¹₂₁ Γ¹₂₂



=







0 f f⁰ f² f f⁰

f² 0





 and Γ²₁₁ Γ²₁₂ Γ²₂₁ Γ²₂₂



=









− f f⁰

(f⁰)²+ (g⁰)² 0 0 f⁰f⁰⁰+ g⁰g⁰⁰

(f⁰)²+ (g⁰)²







 this implies that X(u₁(t), u₂(t)) is a geodesic of the surface of revolution S if u₁, u₂ satisfy the system ofequations

u⁰⁰₁ +2f f⁰

f² u⁰₁u⁰₂ = 0 and u⁰⁰₂ − f f⁰

(f⁰)²+ (g⁰)²(u⁰₁)²+ f⁰f⁰⁰+ g⁰g⁰⁰

(f⁰)²+ (g⁰)²(u⁰₂)² = 0, (††)

where u⁰_k = du_k

dt , f⁰ = df

du₂ and g⁰ = dg du₂.

If the meridian γ(s) = {X(u₁, u₂) | u₁ = constant, u₂ = u₂(s)} is parametrized by arc length s, then the 1^st equation of (††)holds, and, since γ⁰(s) = X₁u⁰₁+ X₂u⁰₂ = X₂u⁰₂, 1 = hγ⁰(s), γ⁰(s)i = I_p(γ⁰(s)) = hX₁u⁰₁+X₂u⁰₂, X₁u⁰₁+X₂u⁰₂i = g₂₂(u⁰₂)² =(f⁰)²+(g⁰)²(u⁰₂)², we have

(u⁰₂)^{2 (∗)}= 1

(f⁰)²+ (g⁰)² =⇒ u⁰₂u⁰⁰₂ = − f⁰f⁰⁰+ g⁰g⁰⁰

(f⁰)²+ (g⁰)²2 u⁰₂

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

dsf⁰ = f⁰⁰u⁰₂ and d

dsg⁰ = g⁰⁰u⁰₂. Multiplying both sides by u⁰₂, we get (u⁰₂)²u⁰⁰₂ = − f⁰f⁰⁰+ g⁰g⁰⁰

(f⁰)²+ (g⁰)²2 (u⁰₂)² =⇒^(∗) u⁰⁰₂ = −f⁰f⁰⁰+ g⁰g⁰⁰

(f⁰)²+ (g⁰)²(u⁰₂)² the 2^nd equation of (††) and this implies that arc length parametrized meridians are geodesics.

If the parallel γ(s) = {X(u₁, u₂) | u₂ = constant, u₁ = u₁(s)} is parametrized by arc length s, since 1 = Ip(γ⁰(s)) = (f (u2))²(u⁰₁)², we have (u⁰₁)² = 1/f (u2) = constant 6= 0 which implies that 2u⁰₁u⁰⁰₁ = 0 =⇒ u⁰⁰₁ = 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 f⁰

(f⁰)²+ (g⁰)²(u⁰₁)² = 0 =⇒ f⁰ = 0 since f 6= 0, u⁰₁ 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 α⁰(s) at p,  Dα⁰(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.