14.4. Normal and the Second Fun- damental Form Normal Curvature . . . Geodesic Curvature . . . In this section, we take a closer look at the curvature at a point of a curve C on a X.

Assuming that C is parameterized by arc length, we will see Home Page that the vector X00(s) (which is equal to κ−→n , where −→n is the Title Page principal normal to the curve C at p, and κ is the curvature) can be written as JJ II

−→ −→ J I κ n = κN N + κg ng , Page 683 of 711 −→ where N is the normal to the surface at p, and κg ng is a Go Back tangential component normal to the curve. Full Screen

The component κN is called the normal curvature. Close

Quit Computing it will lead to the second fundamental form, an- other very important quadratic form associated with a surface. The component κg is called the geodesic curvature. Normal Curvature . . . It turns out that it only depends on the first fundamental form, Geodesic Curvature . . . but computing it is quite complicated, and this will lead to the Christoffel symbols.

3 3 Let f:]a, b[→ E be a curve, where f is a least C -continuous, Home Page and assume that the curve is parameterized by arc length. Title Page

We saw in Chapter 13, section 13.6, that if f 0(s) 6= 0 and JJ II 00 f (s) 6= 0 for all s ∈]a, b[ (i.e., f is biregular), we can associate J I −→ −→ −→ to the point f(s) an orthonormal frame ( t , n , b ) called the Page 684 of 711 Frenet frame, where Go Back

−→ Full Screen t = f 0(s), f 00(s) Close −→n = , kf 00(s)k Quit −→ −→ b = t × −→n . −→ The vector t is the unit vector, the vector −→n is −→ Normal Curvature . . . called the principal normal, and the vector b is called the Geodesic Curvature . . . binormal.

Furthermore the curvature κ at s is κ = kf 00(s)k, and thus,

Home Page f 00(s) = κ−→n . Title Page −→ The principal normal n is contained in the osculating JJ II 0 00 at s, which is just the plane spanned by f (s) and f (s). J I

Page 685 of 711 Recall that since f is parameterized by arc length, the vector f 0(s) is a , and thus Go Back

Full Screen f 0(s) · f 00(s) = 0, Close which shows that f 0(s) and f 00(s) are linearly independent and Quit orthogonal, provided that f 0(s) 6= 0 and f 00(s) 6= 0. Now, if C: t 7→ X(u(t), v(t)) is a curve on a surface X, as- suming that C is parameterized by arc length, which implies Normal Curvature . . . that Geodesic Curvature . . . (s0)2 = E(u0)2 + 2F u0v0 + G(v0)2 = 1, we have 0 0 0 X (s) = Xuu + Xvv , Home Page X00(s) = κ−→n , Title Page −→ 0 0 and t = Xuu + Xvv is indeed a unit tangent vector to the JJ II curve and to the surface, but −→n is the principal normal to the curve, and thus it is not necessarily orthogonal to the tangent J I Page 686 of 711 plane Tp(X) at p = X(u(t), v(t)).

Go Back Thus, if we intend to study how the curvature κ varies as Full Screen the curve C passing through p changes, the Frenet frame −→ −→ Close ( t , −→n , b ) associated with the curve C is not really adequate, −→ Quit since both −→n and b will vary with C (and −→n is undefined when κ = 0). Thus, it is better to pick a frame associated with the normal −→ −→ Normal Curvature . . . to the surface at p, and we pick the frame ( t , ng , N) defined as follows.: Geodesic Curvature . . . Definition 14.4.1 Given a surface X, given any curve C: t 7→ X(u(t), v(t)) on X, for any point p on X, the orthonor- −→ −→ mal frame ( t , ng , N) is defined such that Home Page

−→ 0 0 Title Page t = Xuu + Xvv , X × X N = u v , JJ II kXu × Xvk J I −→ −→ ng = N × t , Page 687 of 711 where N is the normal vector to the surface X at p. The vector Go Back −→ ng is called the geodesic normal vector (for reasons that will Full Screen become clear later). Close

Quit For simplicity of notation, we will often drop arrows above vectors if no confusion may arise. −→ Observe that ng is the unit normal vector to the curve C Normal Curvature . . . contained in the Tp(X) at p. Geodesic Curvature . . . −→ −→ If we use the frame ( t , ng , N), we will see shortly that X00(s) = κ−→n can be written as

−→ −→ Home Page κ n = κN N + κg ng . Title Page −→ The component κN N is the orthogonal projection of κ n onto JJ II the normal direction N, and for this reason κN is called the J I normal curvature of C at p. Page 688 of 711

−→ −→ Go Back The component κg ng is the orthogonal projection of κ n onto the tangent space Tp(X) at p. Full Screen

Close We now show how to compute the normal curvature. This will uncover the second fundamental form. Quit Using the abbreviations Normal Curvature . . . ∂2X ∂2X ∂2X X = ,X = ,X = , Geodesic Curvature . . . uu ∂u2 uv ∂u∂v vv ∂v2

0 0 0 since X = Xuu + Xvv , using the chain rule, we get

00 0 2 0 0 0 2 00 00 Home Page X = Xuu(u ) + 2Xuvu v + Xvv(v ) + Xuu + Xvv . Title Page

In order to decompose X00 = κ−→n into its normal component JJ II (along N) and its tangential component, we use a neat trick J I suggested by Eugenio Calabi. Page 689 of 711

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Quit Recall that Normal Curvature . . . (−→u × −→v ) × −→w = (−→u · −→w )−→v − (−→w · −→v )−→u . Geodesic Curvature . . .

Using this identity, we have 0 2 0 0 0 2 (N × (Xuu(u ) + 2Xuvu v + Xvv(v ) ) × N 0 2 0 0 0 2 Home Page = (N · N)(Xuu(u ) + 2Xuvu v + Xvv(v ) ) 0 2 0 0 0 2 Title Page − (N · (Xuu(u ) + 2Xuvu v + Xvv(v ) ))N. Since N is a unit vector, we have N·N = 1, and consequently, JJ II since J I

−→ 00 0 2 0 0 0 2 00 00 Page 690 of 711 κ n = X = Xuu(u ) + 2Xuvu v + Xvv(v ) + Xuu + Xvv , we can write Go Back

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−→ 0 2 0 0 0 2 κ n = (N · (Xuu(u ) + 2Xuvu v + Xvv(v ) ))N Close 0 2 0 0 0 2 + (N × (Xuu(u ) + 2Xuvu v + Xvv(v ) )) × N Quit 00 00 + Xuu + Xvv . Thus, it is clear that the normal component is Normal Curvature . . .

Geodesic Curvature . . . 0 2 0 0 0 2 κN N = (N · (Xuu(u ) + 2Xuvu v + Xvv(v ) ))N, and the normal curvature is given by Home Page κ = N · (X (u0)2 + 2X u0v0 + X (v0)2). N uu uv vv Title Page

JJ II Letting J I

L = N · Xuu,M = N · Xuv,N = N · Xvv, Page 691 of 711

Go Back we have Full Screen κ = L(u0)2 + 2Mu0v0 + N(v0)2. N Close

Quit It should be noted that some authors (such as do Carmo) use the notation Normal Curvature . . . Geodesic Curvature . . . e = N · Xuu, f = N · Xuv, g = N · Xvv. Recalling that X × X N = u v , kXu × Xvk using the Lagrange identity Home Page (−→u · −→v )2 + k−→u × −→v k2 = k−→u k2k−→v k2, Title Page we see that JJ II J I p 2 kXu × Xvk = EG − F , Page 692 of 711

Go Back and L = N · Xuu can be written as Full Screen

(Xu × Xv) · Xuu (Xu,Xv,Xuu) L = √ = √ , Close EG − F 2 EG − F 2 Quit where (Xu,Xv,Xuu) is the determinant of the three vectors. Some authors (including Gauss himself and Darboux) use the notation Normal Curvature . . . Geodesic Curvature . . .

D = (Xu,Xv,Xuu), 0 D = (Xu,Xv,Xuv), D00 = (X ,X ,X ), u v vv Home Page

Title Page and we also have JJ II D D0 D00 L = √ ,M = √ ,N = √ . J I 2 2 2 EG − F EG − F EG − F Page 693 of 711

Go Back These expressions were used by Gauss to prove his famous Theorema Egregium. Full Screen

Close Since the quadratic form (x, y) 7→ Lx2 + 2Mxy + Ny2 plays Quit a very important role in the theory of surfaces, we introduce the following definition. Normal Curvature . . . Definition 14.4.2 Given a surface X, for any point Geodesic Curvature . . . p = X(u, v) on X, letting

L = N · Xuu,M = N · Xuv,N = N · Xvv, where N is the unit normal at p, the quadratic form (x, y) 7→ 2 2 Lx +2Mxy+Ny is called the second fundamental form of X Home Page at p. It is often denoted as IIp. For a curve C on the surface Title Page X (parameterized by arc length), the quantity κN given by the formula JJ II 0 2 0 0 0 2 J I κN = L(u ) + 2Mu v + N(v ) Page 694 of 711 is called the normal curvature of C at p. Go Back

The second fundamental form was introduced by Gauss in Full Screen

1827. Close

Quit Unlike the first fundamental form, the second fundamental form is not necessarily positive or definite. Properties of the surface expressible in terms of the first fun- damental are called intrinsic properties of the surface X. Normal Curvature . . . Geodesic Curvature . . . Properties of the surface expressible in terms of the second fundamental form are called extrinsic properties of the surface X. They have to do with the way the surface is immersed in 3 E . Home Page

Title Page As we shall see later, certain notions that appear to be extrin- sic turn out to be intrinsic, such as the geodesic curvature and JJ II the . J I

Page 695 of 711 This is another testimony to the genius of Gauss (and Bonnet, Christoffel, etc.). Go Back

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Quit Remark: As in the previous section, if X is not injective, the Normal Curvature . . . second fundamental form IIp is not well defined. Again, we will not worry too much about this, or assume X injective. Geodesic Curvature . . .

It should also be mentioned that the fact that the normal curvature is expressed as

Home Page 0 2 0 0 0 2 κN = L(u ) + 2Mu v + N(v ) Title Page has the following immediate corollary known as Meusnier’s theorem (1776). JJ II J I Lemma 14.4.3 All curves on a surface X and having the Page 696 of 711 same tangent at a given point p ∈ X have the same nor- mal curvature at p. Go Back In particular, if we consider the curves obtained by intersecting Full Screen the surface with planes containing the normal at p, curves Close called normal sections, all curves tangent to a normal section Quit at p have the same normal curvature as the normal section. Furthermore, the principal normal of a normal section is collinear Normal Curvature . . . with the normal to the surface, and thus, |κ|=|κN |, where κ Geodesic Curvature . . . is the curvature of the normal section, and κN is the normal curvature of the normal section.

We will see in a later section how the curvature of normal sections varies. Home Page

Title Page We can easily give an expression for κN for an arbitrary pa- rameterization. JJ II

J I Indeed, remember that Page 697 of 711 !2 ds = kC˙ k2 = E u˙ 2 + 2F u˙v˙ + G v˙ 2, Go Back dt Full Screen and by the chain rule Close du du dt u0 = = , Quit ds dt ds and since a change of parameter is a diffeomorphism, we get Normal Curvature . . . 0 u˙ u = ! Geodesic Curvature . . . ds dt and from 0 2 0 0 0 2 Home Page κN = L(u ) + 2Mu v + N(v ) , Title Page we get JJ II Lu˙ 2 + 2Mu˙v˙ + Nv˙ 2 κ = . J I N Eu˙ 2 + 2F u˙v˙ + Gv˙ 2 Page 698 of 711

It is remarkable that this expression of the normal curvature Go Back uses both the first and the second fundamental form! Full Screen

00 00 Close We still need to compute the tangential part Xt of X . Quit We found that the tangential part of X00 is Normal Curvature . . .

Geodesic Curvature . . . 00 0 2 0 0 0 2 Xt = (N × (Xuu(u ) + 2Xuvu v + Xvv(v ) )) × N 00 00 + Xuu + Xvv .

Home Page This vector is clearly in the tangent space Tp(X) (since the first part is orthogonal to N, which is orthogonal to the tangent Title Page space). JJ II

Furthermore, X00 is orthogonal to X0 (since X0 · X0 = 1), J I 00 00 −→ 0 Page 699 of 711 and by dotting X = κN N + Xt with t = X , since the −→ 00 −→ Go Back component κN N · t is zero, we have Xt · t = 0, and thus −→ 00 Full Screen Xt is also orthogonal to t , which means that it is collinear −→ −→ with ng = N × t . Close

Quit Therefore, we showed that Normal Curvature . . . −→ −→ κ n = κN N + κg ng , Geodesic Curvature . . . where 0 2 0 0 0 2 κN = L(u ) + 2Mu v + N(v ) and Home Page

Title Page −→ 0 2 0 0 0 2 κg ng = (N × (Xuu(u ) + 2Xuvu v + Xvv(v ) )) × N JJ II 00 00 +Xuu + Xvv . J I

−→ Page 700 of 711 The term κg ng is worth an official definition. Go Back

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Quit Normal Curvature . . .

Definition 14.4.4 Given a surface X, given any curve Geodesic Curvature . . . C: t 7→ X(u(t), v(t)) on X, for any point p on X, the quantity κg appearing in the expression −→ −→ κ n = κN N + κg ng Home Page giving the acceleration vector of X at p is called the geodesic Title Page curvature of C at p. JJ II

−→ J I In the next section, we give an expression for κg ng in terms Page 701 of 711 of the basis (Xu,Xv).

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Quit 14.5. Geodesic Curvature and the Christof- fel Symbols Normal Curvature . . . Geodesic Curvature . . . We showed that the tangential part of the curvature of a curve −→ C on a surface is of the form κg ng .

We now show that κn can be computed only in terms of the Home Page first fundamental form of X, a result first proved by Ossian Title Page Bonnet circa 1848. JJ II

The computation is a bit involved, and it will lead us to the J I Christoffel symbols, introduced in 1869. Page 702 of 711

−→ Go Back Since ng is in the tangent space Tp(X), and since (Xu,Xv) is a basis of Tp(X), we can write Full Screen

−→ Close κg ng = AXu + BXv, Quit form some A, B ∈ R. However, Normal Curvature . . .

−→ −→ Geodesic Curvature . . . κ n = κN N + κg ng , and since N is normal to the tangent space, N · Xu = N · Xv = 0, and by dotting −→ Home Page κg ng = AXu + BXv Title Page with Xu and Xv, since E = Xu · Xu, F = Xu · Xv, and G = JJ II Xv · Xv, we get the equations: J I

−→ Page 703 of 711 κ n · Xu = EA + FB, −→ κ n · Xv = FA + GB. Go Back

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On the other hand, Close

Quit −→ 00 00 00 0 2 0 0 0 2 κ n = X = Xuu + Xvv + Xuu(u ) + 2Xuvu v + Xvv(v ) . Dotting with Xu and Xv, we get Normal Curvature . . .

Geodesic Curvature . . . −→ 00 00 0 2 κ n · Xu = Eu + F v + (Xuu · Xu)(u ) 0 0 0 2 + 2(Xuv · Xu)u v + (Xvv · Xu)(v ) , −→ 00 00 0 2 κ n · Xv = F u + Gv + (Xuu · Xv)(u ) 0 0 0 2 Home Page + 2(Xuv · Xv)u v + (Xvv · Xv)(v ) .

Title Page

At this point, it is useful to introduce the Christoffel symbols JJ II

(of the first kind) [α β; γ], defined such that J I

Page 704 of 711 [α β; γ] = Xαβ · Xγ, Go Back

Full Screen where α, β, γ ∈ {u, v}. It is also more convenient to let u = u1 and v = u2, and to denote [uα vβ; uγ] as [α β; γ]. Close

Quit Doing so, and remembering that Normal Curvature . . . −→ κ n · Xu = EA + FB, Geodesic Curvature . . . −→ κ n · Xv = FA + GB, we have the following equation:     EF A Home Page = FG B Title Page  EF   u00  X  [α β; 1] u0 u0  1 + α β . FG u00 [α β; 2] u0 u0 JJ II 2 α=1,2 α β β=1,2 J I

Page 705 of 711 However, since the first fundamental form is positive definite, Go Back EG − F 2 > 0, and we have Full Screen  −1   EF G −F Close = (EG − F 2)−1 . FG −FE Quit Thus, we get Normal Curvature . . .

Geodesic Curvature . . .    00  A u1 = 00 B u2 X  G −F   [α β; 1] u0 u0  + (EG − F 2)−1 α β . −FE [α β; 2] u0 u0 α=1,2 α β Home Page β=1,2

Title Page

It is natural to introduce the Christoffel symbols (of the second JJ II k kind) Γi j, defined such that J I

 1      Page 706 of 711 Γi j 2 −1 G −F [i j; 1] 2 = (EG − F ) . Go Back Γi j −FE [i j; 2]

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Quit Finally, we get Normal Curvature . . . 00 X 1 0 0 A = u1 + Γi j uiuj, Geodesic Curvature . . . i=1,2 j=1,2 00 X 2 0 0 B = u2 + Γi j uiuj, i=1,2 j=1,2 Home Page and Title Page −→ κg ng =     JJ II J I 00 X 1 0 0 00 X 2 0 0 u1 + Γi j uiuj Xu + u2 + Γi j uiuj Xv.     Page 707 of 711 i=1,2 i=1,2 j=1,2 j=1,2 Go Back

Full Screen We summarize all the above in the following lemma. Close

Quit Normal Curvature . . . Lemma 14.5.1 Given a surface X and a curve C on X, for Geodesic Curvature . . . any point p on C, the tangential part of the curvature at p is given by −→ κg ng =     Home Page X X u00 + Γ1 u0u0  X + u00 + Γ2 u0u0  X ,  1 i j i j u  2 i j i j v Title Page i=1,2 i=1,2 j=1,2 j=1,2 JJ II k where the Christoffel symbols Γi j are defined such that J I −1  1      Page 708 of 711 Γi j EF [i j; 1] 2 = , Γi j FG [i j; 2] Go Back and the Christoffel symbols [i j; k] are defined such that Full Screen

[i j; k] = Xij · Xk. Close

Note that Quit [i j; k] = [j i; k] = Xij · Xk. Looking at the formulae Normal Curvature . . .

[α β; γ] = Xαβ · Xγ Geodesic Curvature . . . for the Christoffel symbols [α β; γ], it does not seem that these symbols only depend on the first fundamental form, but in fact they do! Home Page

After some calculations, we have the following formulae show- Title Page ing that the Christoffel symbols only depend on the first fun- damental form: JJ II 1 1 J I [1 1; 1] = E , [1 1; 2] = F − E , 2 u u 2 v Page 709 of 711 1 1 [1 2; 1] = E , [1 2; 2] = G , Go Back 2 v 2 u 1 1 Full Screen [2 1; 1] = Ev, [2 1; 2] = Gu, 2 2 Close 1 1 [2 2; 1] = F − G , [2 2; 2] = G . Quit v 2 u 2 v Another way to compute the Christoffel symbols [α β; γ], is to proceed as follows. For this computation, it is more con- Normal Curvature . . . Geodesic Curvature . . . venient to assume that u = u1 and v = u2, and that the first fundamental form is expressed by the matrix  g g   EF  11 12 = , g21 g22 FG Home Page where gαβ = Xα · Xβ. Let Title Page ∂gαβ gαβ|γ = . ∂uγ JJ II Then, we have J I

Page 710 of 711 ∂gαβ gαβ|γ = = Xαγ · Xβ + Xα · Xβγ = [α γ; β] + [β γ; α]. ∂uγ Go Back

From this, we also have Full Screen

gβγ|α = [α β; γ] + [α γ; β], Close and Quit gαγ|β = [α β; γ] + [β γ; α]. From all this, we get Normal Curvature . . .

2[α β; γ] = gαγ|β + gβγ|α − gαβ|γ. Geodesic Curvature . . .

γ As before, the Christoffel symbols [α β; γ] and Γα β are related via the Riemannian metric by the equations Home Page  −1 γ g11 g12 Title Page Γα β = [α β; γ]. g21 g22 JJ II

This seemingly bizarre approach has the advantage to gener- J I alize to Riemannian . In the next section, we study Page 711 of 711 the variation of the normal curvature. Go Back

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