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Chapter 1 DIFFERENTIAL GEOMETRY of REAL MANIFOLDS Chapter 1 DIFFERENTIAL GEOMETRY OF REAL MANIFOLDS 1.1 Simplest Applications of Structure Equations 1.1.1 Moving Frames in Euclidean Spaces We equip RN with the standard inner product <, >. By a moving frame in U ⊆ RN we mean a choice of orthonormal bases {e1(x), ··· , eN (x)} for all TxU, x ∈ U. Taking exterior derivatives we obtain X X dx = ωAeA, deA = ωBAeB (1.1.1) A B where ωA’s and ωAB’s are 1-forms. Since ωA and ωAB depend on the point x and the choice of the moving frame {e1, ··· , eN }, their natural domain of definition is the principal bundle Fg → U of orthonormal frames on U. However, due to the functorial property of the exterior derivative (f ?(dη) = df ?(η)), the actual domain is immaterial for many calculations and sometimes we use local parametrizations in our computations. The orthonormality condition implies 0 = d < eA, eB >=< deA, eB > + < eA, deB > and consequently ωAB + ωBA = 0. (1.1.2) That is, the matrix valued 1-form ω = (ωAB) takes values in the Lie algebra SO(N). From ddx = 0 and ddeA = 0 we obtain X X dωA + ωAB ∧ ωB = 0; dωAB + ωAC ∧ ωCB = 0. (1.1.3) B C 195 196 CHAPTER 1. DIFFERENTIAL GEOMETRY ... These equations are often called the structure equations for Euclidean space or more precisely for the group of rigid motions of Euclidean space. The second set of equations is also known as the structure equations for the (proper) orthogonal group. These equations are a special case of Maurer-Cartan equations as discussed in chapter 1, and here we have given another derivation of them. In fact, by fixing an origin and an orthonormal frame the set of (positively oriented) frames on Rn can be identified with the group of (proper) Euclidean motions, and (1.1.3) becomes identical with the Maurer-Cartan equations where we have represented the (N + 1) × (N + 1) matrix U −1dU in the form (see chapter 1, §3.5) ω11 ··· ω1N ω1 . .. . ωN1 ··· ωNN ωN 0 ··· 0 0 1.1.2 Curves in the Plane Geometry of curves in the plane is the simplest and oldest area of differential geometry. Let us interpret the 1-form ω12 in the context of plane curves. Choose the orthonormal moving frame e1, e2 such that e1 is the unit tangent vector field to γ and e1, e2 is positively oriented for the standard orientation of the plane. Then we have de1 = ω21e2 and de2 = ω12e1. The 1-form ω21 or ω21(e1) has a familiar geometric interpretation. Set γ(t) = (x(t), y(t)) and let dx x˙ = dt etc. Then x˙y¨ − x¨y˙ ω = dt = dφ, (1.1.4) 21 x˙ 2 +y ˙2 y˙ where tan φ = x˙ , i.e., tan φ is the slope of the tangent to γ at γ(t). Therefore ω21(e1) is the curvature κ of the plane curve as defined in elementary calculus. (This interpretation of ω21 may be somewhat misleading in higher dimensions as we shall see later.) The quantity |ω21(e1)| is independent of the parametrization of γ although the sign of ω21(e1) depends on the choice of orientation for R2 and whether we are traversing the curve in the counterclock- wise or the clockwise direction. Since it is conventional to assign positive curvature to the circle, we use the standard orientation for R2 and move counterclockwise along the curve. It is also convenient to parametrize γ by its arc-length s, i.e. t = s, so thatx ˙ 2 +y ˙2 = 1. We shall do so for the remainder of this subsection. We also recall from our treatment of immersions of the circle into R2 (chapter 1, §5.5 (????)) that if G : C → S1 ⊂ C is defined by G(s) = ei(s), i = 1 or 2, and dθ denotes the standard measure on the circle, then G?(dθ) = κ(s)ds. (1.1.5) R Therefore C ω12 is the winding number of the curve C. 1.1. SIMPLEST APPLICATIONS ... 197 Example 1.1.1 Let γ : I → R2 be a curve with curvature κ and assume γ is parametrized by arc-length s. We give a geometric interpretation of curvature of Γ which is sometimes useful. Assume κ 6= 0 and by reversing the orientation of γ we assume κ > 0. Let e1, e2 be moving frame with e1 (resp. e2) tangent (resp. normal) to the curve. Consider the mapping Φ: I × [0, ) → R2 Φ(s, t) = γ(s) + te2(s). 1 From calculus we know that if < mins |κ(s)| , then the mapping Φ is injective. Furthermore, the image of (0, 1) × (0, ) is an open subset of R2. A general point in U = Φ((0, 1) × (0, )) has a unique representation q = γ(s) + te2 and dq = [ds + tω21]e1(s) + dte2(s). We obtain dvU = ds ∧ dt + tω21 ∧ dt for the volume element on U ⊂ R2. This expression for the volume element is no longer valid if is large. In fact, if κ(s) > 0 (for the chosen orientation of the curve), and the point y 1 0 along the normal e3 is at a distance > κ(s) from γ(s), then there are points s 6= s arbitrarily close to s such that d(y, γ(s)) ≥ d(y, γ(s0)), (1.1.6) and the map Φ is no longer a diffeomorphism. This observation will be useful later. ♠ Let γ : [0,L] → R2 be a simple closed curve parametrized by arc length s and denote its image by Γ. Such a curve decomposes the plane into two connected components which are the interior Γi and the exterior Γe of Γ. This is a topological fact which is so familiar from experience that assuming its validity will not be a cause for concern. For a C1 simple closed curve, we can define the interior Γi as the set of points q 6∈ Γ such that a generic ray (i.e., half infinite straight line) starting at q, intersects Γ is an odd number of points. Later, we will discuss a generalization and rigorous proof of this fact. We say Γ is convex if Γi is a convex set. It is not difficult to show that the convexity of Γ is equivalent to any one of the following conditions: 1. For every tangent line T to Γ, Γi ∩ T = ∅. 2. For every tangent line T ,Γi lies on one side of T . 3. The intersection of any straight line with Γ has at most two connected components. 198 CHAPTER 1. DIFFERENTIAL GEOMETRY ... These descriptions of convexity in the plane are so familiar that if necesary we will make use of them without further ado. There is also a differential geometric condition which implies convexity. Intuitively, this condition says that if the curvature of Γ does not vanish, then the angle that the inward (or outward) normals to Γ make with a fixed direction, e.g. the positive x-axis gives a parametrization of the curve Γ and Γ is convex, in other words, Lemma 1.1.1 The Gauss map G :Γ → S1 of a simple closed curve with nowhere vanishing curvature is a diffeomorphism, and Γ is convex. Proof - Since the winding number of a simple closed curve is 2π (see chapter 1, §5.5) we 0 know that G is onto. If G(s1) = G(s2) then by Rolle’s theorem G (t) vanishes for some t and consequently there is a point with zero curvature. Similarly, if Γ does not lie on one side of a tangent line T , then it intersects Γ in at least two distinct points q1, q2. By Rolle’s theorem there is a point between q1 and q2 where the tangent line is parallel to T . This contradicts the first assertion of the lemma proving convexity of Γ. ♣ It is customary to refer to a simple closed curve Γ ⊂ R2 with nowhere vanishing curvature as strictly convex. Since lemma 1.1.1 shows that the unit circle parametrizes a simple closed strictly convex curve, we ask whether any positive function κ on S1 can be realized as the curvature of such a curve Γ with κ(θ) the curvature at the unique point on Γ with normal iθ e2 = e . The following proposition shows that there is a necessary condition to be satisfied: Proposition 1.1.1 Let e1, e2 denote the unit tangent and normal to the simple closed curve Γ given by an embedding γ : S1 → R2, and κ˜ denote the curvature of Γ regarded as a function on S1. Then Z 1 Z 1 e2dθ = 0 or equivalently e1dθ = 0. S1 κ˜ S1 κ˜ If κ˜ is positive, then this condition is also sufficient for the existence of a strictly convex simple closed curve with curvature κ˜ relative to the parametrization of lemma 1.1.1 Proof - Since γ is a simple closed curve, it can be given as an embedding of S1 into R2. Let s be the arc length along Γ and θ the parameter along S1. Then dγ dγ ds = . (1.1.7) dθ ds dθ 1 R Therefore dγ = κ˜ e1dθ and the necessity follows by integration S1 dγ = 0 and the fact that π e1 and e2 differ by the constant rotation through 2 . To prove the sufficiency assertion we integrate the equation dγ 1 i( π +θ) = e , where e = e 2 , dθ κ˜ 1 1 1.1. SIMPLEST APPLICATIONS ... 199 on the circle S1. The periodicity of the solution γ follows from the hypothesis. From (1.1.7) we see thatκ ˜ is the curvature of and e1 is the unit tangent vector field to γ.
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