LectureNotes12U

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Math 497C Nov 11, 20041

Curves and Surfaces

Fall 2004, PSU

Lecture Notes 12

2.6 Gauss’s formulas, and Christoffel Symbols

Let X : U → R3 be a proper regular patch for a surface M , and set X i :=DiX . Then

{X 1, X 2, N }may be regarded as a moving bases of frame for R3 similar to the FrenetSerret frames for curves. We should emphasize, however, two importantdifferences: (i) there is no canonical choice of a moving bases for a surfaceor a piece of surface ({X 1, X 2, N } depends on the choice of the chart X );(ii) in general it is not possible to choose a patch X so that {X 1, X 2, N } isorthonormal (unless the Gaussian curvature of M vanishes everywhere).

The following equations, the first of which is known as Gauss’s formulas,may be regarded as the analog of Frenet-Serret formulas for surfaces:

X ij =

2k=1

ΓkijX k + lijN, and N i = −

2 j=1

l jiX j.

The coefficients Γkij are known as the Christoffel symbols, and will be deter-

mined below. Recall that lij are just the coefficients of the second fundamen-tal form. To find out what l ji are note that

−lik = −N, X ik = N i, X k = −2

j=1

l ji X j, X k = −

2 j=1

l ji g jk .

Thus (lij) = (l ji )(gij). So if we let (gij) := (gij)−1, then (l ji ) = (lij)(gij), which

yields

l ji =

2k=1

likgkj .

1Last revised: November 12, 2004

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2.7 The Gauss and Codazzi-Mainardi Equations, Rie-

mann Curvature Tensor, and a Second Proof of Gauss’s Theorema Egregium

Here we shall derive some relations between lij and gij. Our point of depar-ture is the simple observation that if X : U → R3 is a C 3 regular patch, then,since partial derivatives commute,

X ijk = X ikj .

Note that

X ijk = 2

l=1

ΓlijX l + lijN

k

=2

l=1

(Γlij)kX l +

2l=1

ΓlijX lk + (lij)kN + lijN k

=2

l=1

(Γlij)kX l +

2l=1

Γlij

2m=1

ΓmlkX m + llkN

+ (lij)kN − lij

2l=1

llkX l

=2

l=1

(Γlij)kX l +

2l=1

2m=1

ΓlijΓm

lkX m +2

l=1

Γlij llkN + (lij)kN −

2l=1

lijllkX l

=

2l=1

(Γ

lij)k +

2 p=1

Γ pijΓ

l pk − lij l

lkX l +

2l=1

Γlij llk + (lij)k

N.

Switching k and j yields,

X ikj =2

l=1

(Γl

ik) j +2

p=1

Γ pikΓl

pj − likll j

X l +

2l=1

Γlikllj + (lik) j

N.

Setting the normal and tangential components of the last two equations equalto each other we obtain

(Γlij)k +

2 p=1

Γ pijΓl pk − lijllk = (Γlik) j +

2 p=1

Γ pikΓl pj − likll j,

2l=1

Γlij llk + (lij)k =

2l=1

Γlikllj + (lik) j .

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These equations may be rewritten as

(Γlik) j − (Γl

ij)k +2

p=1

Γ p

ikΓl pj − Γ p

ijΓl pk

= likl

l j − lij l

lk, (Gauss)

2l=1

Γlikllj − Γl

ij llk

= (lij)k − (lik) j, (Codazzi-Mainardi)

and are known as the Gauss’s equations and the Codazzi-Mainardi equations

respectively. If we define the Riemann curvature tensor as

Rlijk := (Γl

ik) j − (Γlij)k +

2

p=1

Γ pikΓl

pj − Γ pijΓl

pk,then Gauss’s equation may be rewritten as

Rlijk = likl

l j − lij l

lk.

Now note that

2l=1

Rlijkglm = lik

2l=1

ll jglm − lij

2l=1

llkglm = likl jm − lijlkm.

In particular, if i = k = 1 and j = m = 2, then

2l=1

Rl121gl2 = l11l22 − l12l21 = det(lij) = K det(gij).

So it follows that

K =R1

121g12 + R2

121g22

det(gij),

which shows that K is intrinsic and gives another proof of Gauss’s TheoremaEgregium.

Exercise 4. Show that if M = R2, hen Rlijk = 0 for all 1 ≤ i, l ,j ,k ≤ 2

both intrinsically and extrinsically.

Exercise 5. Show that (i) Rlijk = −Rl

ikj , hence Rlijj = 0, and (ii) Rl

ijk +Rl jki + Rl

kij ≡ 0.

Exercise 6. Compute the Riemann curvature tensor for S2 both intrinsi-cally and extrinsically.

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2.8 Fundamental Theorem of Surfaces

In the previous section we showed that if gij and lij are the coefficients of thefirst and second fundamental form of a patch X : U → M , then they mustsatisfy the Gauss and Codazzi-Maindardi equations. These conditions turnout to be not only necessary but also sufficient in the following sense.

Theorem 7 (Fundamental Theorem of Surfaces). Let U ⊂ R2 be an

open neighborhood of the origin (0, 0), and gij : U → R, lij : U → R be

differentiable functions for i, j = 1, 2. Suppose that gij = g ji, lij = l ji,

g11 > 0, g22 > 0 and det(gij) > 0. Further suppose that gij and lij satisfy

the Gauss and Codazzi-Mainardi equations. Then there exists and open set

V ⊂ U , with (0, 0) ∈ V and a regular patch X : V → R with gij and lij as

its first and second fundamental forms respectively. Further, if Y : V → R3

is another regular patch with first and second fundamental forms gij and lij,

then Y differs from X by a rigid motion.

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