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Intrinsic Geometry

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Intrinsic Geometry Chapter 3 Surfaces: intrinsic geometry It is the geometry of objects associated to the surface which depend only on the first fundamental form. Obvious examples are lengths of curves, angles between tangent vectors and areas of regions in the the surface. Less obvious examples are geodesics (yet to be defined) and the Gaussian curvature (Theo- rema Egregium). We will first discuss local questions. 3.1 Directional derivative We start by recalling the concept of directional derivative of vector fields on Rn. Let Y : W Rn be a smooth vector field defined on an open subset W of n n → R , let v R be a fixed vector and p W . Then the directional derivative of Y ∈ ∈ along v at p is Y (p + tv) Y (p) DvY p = dYp(v) = lim − . t 0 | → t n Note that if γ : ( ǫ,ǫ) R is a smooth curve with γ(0) = p and γ′(0) = v, − → then Y (γ(t)) Y (γ(0)) d lim − = Y (γ(t)) t 0 t dt t=0 → = dY γ(0)(γ′(0)) (by the chain rule) = dYp(v) = DvY p, | in other words, DvY p depends only on the values of Y along a smooth curve | through p with velocity v. n As a particular case, consider the canonical basis e1,...,en of R and n { } denote by (x1,...,xn) the standard coordinates in R . Then ∂Y De Y p = i (p) i | ∂x 31 32 CHAPTER 3. SURFACES: INTRINSIC GEOMETRY i and, if v = i v ei, then i ∂Y DvY = v . P ∂xi i X A word about notation: If X : W Rn is another smooth vector field, then n → X(p) is a vector in R and we write DX Y p = D Y p. | X(p) | 3.2 Vector fields on surfaces Let S be a surface in R3, and let V S be an open subset. A (smooth) vector ⊂ 3 field on V is a (smooth) map X : V R . We say that X is tangent to S (resp. → normal to S) if X(p) TpS (resp. X(p) TpS) for p V . 2 ∈ ⊥ ∈ Let ϕ : U R ϕ(U)= V S be a parametrization. In the following, we ⊂ → ⊂ 2 use (u1,u2) to denote coordinates on the parameter plane R and (x1, x2, x3) to denote coordinates in the ambient R3. According to the above definition, ∂ϕ 1 ∂ϕ 1 3 1 ϕ− , 2 ϕ− : V R ∂u ◦ ∂u ◦ → are vector fields on V tangent to S. For the sake of convenience, henceforth ∂ϕ ∂ϕ we will abuse terminology and say that ∂u1 , ∂u2 are vector fields tangent to S. Note then that any vector field X on V tangent to S can be written as a linear combination 1 ∂ϕ 2 ∂ϕ X = a ∂u1 + a ∂u2 where a1, a2 : V R. It is an easy exercise to check that X is smooth if and → only if a1, a2 are smooth functions (do it!). Similarly, any smooth vector field on V normal to S is of the form ∂ϕ ∂ϕ X = b 1 2 ∂u × ∂u for a smooth function b : V R. → 3.3 Covariant derivative In this section we explain how to differentiate a vector field tangent to a surface along another tangent vector field to obtain a third tangent vector field. Let S be a surface in R3, V S an open subset, p V . Consider vector fields X, Y on ⊂ ∈ V such that X is tangent to S. Our previous discussion about the directional d derivative shows that DX Y p is well defined, namely, it equals t Y (γ(t)) | dt | =0 where γ : ( ǫ,ǫ) S is smooth and γ(0) = p, γ′(0) = v. The association − → p DX Y p 7→ | is a vector field on V . As a special case, if ϕ : U V is a parametrization of S → and ϕ(u)= p for some u U, ∈ ∂(Y ϕ) D ∂ϕ Y p = ∂u◦i (u). ∂ui | 3.3. COVARIANT DERIVATIVE 33 ∂ϕ Further, if Y = ∂uj , ∂ϕ ∂2ϕ D ∂ϕ i p = i j (u). ∂ui ∂u | ∂u u Back to the general case, we now assume that both X and Y are tangent to S. The covariant derivative of Y along X at p is X Y p = (DX Y p)⊤ ∇ | | where ( )⊤ denotes the orthogonal projection onto TpS. The symbol “ ” is · ∇ read “nabla”. In this way, the association p X Y p 7→ ∇ | is a vector field on V tangent to S. Lemma 3.1 (Properties of D and ) Let X, X˜, Y , Y˜ be vector fields tangent to S ∇ and let f be a smooth function on S. Then: 1. ˜ Y = f X Y + ˜ Y ; ∇fX+X ∇ ∇X 2. X (Y + Y˜ )= X Y + X Y˜ ; ∇ ∇ ∇ 3. X (fY )= df(X)Y + f X Y ; ∇ ∇ 4. X Y, Y˜ = X Y, Y˜ + Y, X Y˜ ; h i h∇ i h ∇ i and the same identities hold for replaced by D. ∇ In this statement, fX denotes the tangent vector field p f(p)X(p), and 7→ Y, Y˜ denotes the scalar function p Y (p), Y˜ (p) so that X Y, Y˜ (p) denotes h i 7→ h i h i the the directional derivative in the direction of X(p). Proof. We prove (3)and (4)and leave the restasan exercise. Let γ : ( ǫ,ǫ) − → S be a smooth curve with γ(0) = p, γ′(0) = X(p). Then d DX (fY ) p = t (fY )(γ(t)) | dt| =0 d = t f(γ(t))Y (γ(t)) dt| =0 d d = t f(γ(t)) Y (γ(0)) + f(γ(0)) t Y (γ(t)) dt| =0 · dt| =0 = dfp(X(p))Y (p)+ f(p)DX Y p. | Hence X (fY ) p = (DX (fY ))⊤ ∇ | = (df(X)Y + fDX Y )⊤ = df(X)Y ⊤ + f ( X Y )⊤ (orthogonal projection is linear) ∇ = df(X)Y + f X Y (Y is tangent). ∇ 34 CHAPTER 3. SURFACES: INTRINSIC GEOMETRY Next, d X Y, Y˜ (p) = t Y (γ(t)), Y˜ (γ(t)) h i dt| =0h i d d = t Y (γ(t)), Y˜ (γ(0)) + Y (γ(0)), t Y˜ (γ(t)) hdt| =0 i h dt| =0 i = DX Y p, Y˜ (p) + Y (p),DX Y˜ p h | i h | i ⊤ = (DX Y p)⊤ , Y˜ (p) + Y (p), DX Y˜ p (Y , Y˜ are tangent) h | i h | i = X Y p, Y˜ (p) + Y (p), X Y˜ p , h∇ | i h ∇ | i as we wished. Let X, Y be tangent vector fields. Of course, ν is a normal vector field, so Y, ν =0 and Lemma 3.1(4) says that h i DX Y, ν + Y,DX ν =0. h i h i Le A = dν be the Weingarten operator. Then A(X)= dν(X)= DX ν and − − − we have X Y = (DX Y )⊤ ∇ = DX Y DX Y, ν ν − h i normal component = DX Y + Y,D|X ν {zν } h i = DX Y Y,AX ν − h i = DX Y Π(X, Y )ν. − Hence we arrive at the Gauss formula DX Y = X Y + Π(X, Y )ν. ∇ In the remaining of this section, we show that the covariant derivative ∇ of a surface S is an intrinsic object, namely, it is completely determined by the first fundamental form I; in particular, locally isometric surfaces have the same covariant derivative. This is not an obvious assertion in view of the fact that X Y was defined as the orthogonal projection onto the surface of the direc- ∇ 3 tional derivative DX Y , and so the ambient space R was used in this defini- tion. Let X, Y be vector fields on S. Since we are dealing with a local assertion, we can work in the image of a parametrization ϕ of S and write 2 2 i ∂ϕ j ∂ϕ X = a ∂ui , Y = b ∂uj . i=1 i=1 X X 3.3. COVARIANT DERIVATIVE 35 By using Lemma 3.1, we first obtain a local expression for X Y : ∇ i j ∂ϕ X Y = a ∂ϕ b j ∇ ∇ ∂ui ∂u i,j X j i ∂b ∂ϕ j ∂ϕ = a i j + b ∂ϕ j . (3.2) ∂u ∂u ∇ ∂ui ∂u i,j X ∂ϕ Since ∂ϕ ∂uj is tangent to S, we can write ∇ ∂ui ∂ϕ k ∂ϕ ∂ϕ j = Γij k , (3.3) ∇ ∂ui ∂u ∂u k X k for some smooth functions Γij , the so called Christofell symbols (of the second kind). Substituting into (3.2), k i ∂b j k ∂ϕ X Y = a + b Γ , (3.4) ∇ ∂ui ij ∂uk i j X X which shows that depends only on Γk . ∇ { ij } We can also define the Christofell symbols of the first kind by putting ∂ϕ ∂ϕ Γij,k = I ∂ϕ j , k . ∇ ∂ui ∂u ∂u By using (3.3), we have ∂ϕ ∂ϕ Γ = Γℓ I , ij,k ij ∂uℓ ∂uk ℓ X ℓ = Γij gℓk. ℓ X Multiplying through by gkm, where (gkm) denotes the inverse matrix of I = (gij ), we get m km Γij = Γij,kg . k X k Hence Γ depends only on Γij,k and I. { ij } { } In order to complete our argument, we need to show that Γij,k depends k { } only on I. It is important to notice that Γij (and Γij,k) is symmetric with respect to the indices i, j, namely, k k Γij =Γji. This is immediate from ∂ϕ ∂2ϕ ⊤ ∂ϕ j = i j . ∇ ∂ui ∂u ∂u ∂u 36 CHAPTER 3. SURFACES: INTRINSIC GEOMETRY Next, by using Lemma 3.1(4), we write ∂gij ∂ϕ ∂ϕ ∂ϕ ∂ϕ k = ∂ϕ i , j + i ∂ϕ j , ∂u ∇ ∂uk ∂u ∂u ∂u ∇ ∂uk ∂u so ∂g ij =Γ +Γ .
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