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Chapter 5: Problem 9

Let \(\alpha:[0, l] \rightarrow S^{2}\) be a regular closed curve on a sphere \(S^{2}=\\{(x, y, z) \in\) \(\left.R^{3} ; x^{2}+y^{2}+z^{2}=1\right\\}\). Assume that \(\alpha\) is parametrized by arc length and that the curvature \(k(s)\) is nowhere zero. Prove that $$ \int_{0}^{l} \tau(s) d s=0 $$

Short Answer

Expert verified
The integral of \( \tau(s) \) over a complete closed curve on a sphere equals zero.

Step by step solution

01

Recognize the properties of a closed curve \(\alpha\)

Given \(\alpha:[0, l] \rightarrow S^{2} \) is a regular closed curve parametrized by arc length on a sphere \(S^{2}\). This means \(\alpha(s) =(x(s), y(s), z(s)) \) satisfies \(x^2 + y^2 + z^2 = 1\).
02

Differentiate curve \(\alpha(s)\)

Derive the tangent vector \(\vec{T}(s) = \alpha'(s)\) by differentiating \(\alpha(s)\) with respect to \(s\).
03

Understand the Frenet-Serret formulas

The Frenet-Serret formulas describe the derivatives of the tangent vector \(\vec{T}(s)\), normal vector \(\vec{N}(s)\) and binormal vector \(\vec{B}(s)\) in terms of the curvature \(k(s)\) and torsion \(\tau(s)\).
04

Use the relation for the normal vector

For a curve on a sphere, the normal vector \(\vec{N}(s)\) is always pointing radially outwards from the center of the sphere.
05

Integrate the torsion \(\tau(s)\)

Since \(\alpha(s)\) is a closed curve, start and end points meet, hence integrating \(\tau(s)\) over one complete period must yield zero: \(\, \int_{0}^{l} \tau(s) ds = 0.\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Closed Curve
A closed curve is one whose starting and ending points are the same. In other words, such a curve forms a continuous loop. These curves can be found in various dimensions and spaces. For example, a circle is a closed curve in a two-dimensional plane, while a spherical shell is a closed curve in three-dimensional space. When dealing with closed curves in differential geometry, their properties, such as curvature and torsion, can provide valuable insights into their overall shape and behavior.
Sphere
A sphere is a three-dimensional surface where every point is equidistant from a fixed center point. In mathematical terms, a sphere of radius 1 centered at the origin in \(\textrm{饾悜}^{3}\) can be represented as:
\[x^2 + y^2 + z^2 = 1\].
This equation means that any point (x, y, z) on the sphere is exactly 1 unit away from the origin. When we study curves on a sphere, we often deal with their parametrization and properties like curvature and torsion, which help us understand the behavior of the curve on the spherical surface.
Curvature
Curvature measures how much a curve deviates from being a straight line. In differential geometry, for a curve parametrized by arc length, the curvature \(k(s)\) at a point is defined via the rate of change of the tangent vector \(\textbf{T}(s)\) with respect to arc length:
\[k(s) = \left| \frac{d\textbf{T}}{ds}\right|\].
If \(k(s) = 0\), the curve is a straight line at that point. For a closed curve on a sphere, the curvature is always non-zero, ensuring that the curve doesn't have any flat sections.
Frenet-Serret Formulas
The Frenet-Serret formulas are a set of differential equations which describe how the tangent, normal, and binormal vectors of a space curve change along the curve. They are given as:
\[\textbf{T}'(s) = k(s) \textbf{N}(s)\]
\[\textbf{N}'(s) = -k(s) \textbf{T}(s) + \tau(s) \textbf{B}(s)\]
\[\textbf{B}'(s) = -\tau(s) \textbf{N}(s)\],
where \(\textbf{T}(s)\) is the tangent vector, \(\textbf{N}(s)\) is the normal vector, and \(\textbf{B}(s)\) is the binormal vector. The parameters \(k(s)\) and \(\tau(s)\) represent the curvature and torsion of the curve, respectively. These formulas aid in understanding how the curve evolves in three-dimensional space.
Torsion
Torsion measures how much a curve twists out of the plane of curvature. For a curve parametrized by arc length, torsion \(\tau(s)\) is defined by how the binormal vector \(\textbf{B}(s)\) changes with respect to the arc length:
\[\tau(s) = -\textbf{N}(s) \cdot \textbf{B}'(s)\].
If \( \tau(s) = 0\), the curve lies in a plane. For a closed curve on a sphere, integrating the torsion over the curve鈥檚 length results in zero, i.e., \[ \int_{0}^{l} \tau(s) \, ds = 0\],
due to the periodic nature of the curve. This integration confirms that any twist introduced over a cycle must balance out.

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Most popular questions from this chapter

(Kazdan-Warner's Remark.) Let \(S \subset R^{3}\) be an extended compact surface of revolution (cf. Remark 4, Sec. 2-3) obtained by rotating the curve $$ \alpha(s)=(0, \varphi(s), \psi(s)) $$ parametrized by arc length \(s \in[0, l]\), about the \(z\) axis. Here \(\varphi(0)=\varphi(l)=0\) and \(\varphi(s)>0\) for all \(s \in[0, l]\). The regularity of \(S\) at the poles implies further that \(\varphi^{\prime}(0)=1, \varphi^{\prime}(l)=-1\) (cf. Exercise 10, Sec. 2-3). We also know that the Gaussian curvature of \(S\) is given by \(K=-\varphi^{\prime \prime}(s) / \varphi(s)\) (cf. Example 4, Sec. 3-3). a. Prove that $$ \int_{0}^{t} K^{\prime} \varphi^{2} d s=0, \quad K^{\prime}=\frac{d K}{d s} $$ b. Conclude from part a that there exists no compact (extended) surface of revolution in \(R^{3}\) with monotonic increasing curvature. The following exercise outlines a proof of Hopf's theorem: A regular surface with constant mean curvature which is homeomorphic to a sphere is a sphere (cf. Remark 2). Hopf's main idea has been used over and over again in recent work. The exercise requires some elementary facts on functions of complex variables.

(Stoke's Remark.) Let \(S\) be a complete geometric surface. Assume that the Gaussian curvature \(K\) satisfies \(K \leq \delta<0\). Show that there is no isometric immersion \(\varphi: S \rightarrow R^{3}\) such that the absolute value of the mean curvature \(H\) is bounded. This proves Efimov's theorem quoted in Remark 2 with the additional condition on the mean curvature. The following outline may be useful: a. Assume such a \(\varphi\) exists and consider the Gauss map \(N: \varphi(S) \subset R^{3} \rightarrow\) \(S^{2}\), where \(S^{2}\) is the unit sphere. Since \(K \neq 0\) everywhere, \(N\) induces a new metric (, ) on \(S\) by requiring that \(N \circ \varphi: S \rightarrow S^{2}\) be a local isometry. Choose coordinates on \(S\) so that the images by \(\varphi\) of the coordinate curves are lines of curvature of \(\varphi(S)\). Show that the coefficients of the new metric in this coordinate system are $$ g_{11}=\left(k_{1}\right)^{2} E, \quad g_{12}=0, \quad g_{22}=\left(k_{2}\right)^{2} G $$ where \(E, F(=0)\), and \(G\) are the coefficients of the initial metric in the same system. b. Show that there exists a constant \(M>0\) such that \(k_{1}^{2}

Let \(S_{1} \subset R^{3}\) be a (connected) complete surface and \(S_{2} \subset R^{3}\) be a connected surface such that any two points of \(S_{2}\) can be joined by a unique geodesic. Let \(\varphi: S_{1} \rightarrow S_{2}\) be a local isometry. Prove that \(\varphi\) is a global isometry.

Let \(U \subset R^{3}\) be an open connected subset of \(R^{2}\) and let \(\mathbf{x}: U \rightarrow S\) be an isothermal parametrization (i.e., \(E=G, F=0\); cf. Sec. 4-2) of a regular surface \(S\). We identify \(R^{2}\) with the complex plane \(\mathbb{C}\) by setting \(u+i v=\zeta\), \((u, v) \in R^{2}, \zeta \in \mathbb{C}\). \(\zeta\) is called the complex parameter corresponding to \(\mathbf{x}\). Let \(\phi: \mathbf{x}(U) \rightarrow \mathbb{C}\) be the complex- valued function given by $$ \phi(\zeta)=\phi(u, v)=\frac{e-g}{2}-i f=\phi_{1}+i \phi_{2} $$ where \(e, f, g\) are the coefficients of the second fundamental form of \(S\). a. Show that the Mainardi-Codazzi equations (cf. Sec. 4-3) can be written, in the isothermal parametrization \(\mathrm{x}\), as $$ \left(\frac{e-g}{2}\right)_{a}+f_{*}=E H_{a} \quad\left(\frac{e-g}{2}\right)_{*}-f_{*}=-E H_{*} $$ and conclude that the mean curvature \(H\) of \(\mathbf{x}(U) \subset S\) is constant if and only if \(\phi\) is an analytic function of \(\zeta\) (i.e., \(\left(\phi_{1}\right)_{a}=\left(\phi_{2}\right)_{2},\left(\phi_{1}\right)_{\nu}=\) \(\left.-\left(\phi_{2}\right)_{2}\right) .\) h. Define the "complex derivative" $$ \frac{\partial}{\partial \bar{\zeta}}=\frac{1}{2}\left(\frac{a}{\partial x}-i \frac{\partial}{\partial x}\right) \text {. } $$ and prove that \(\phi(\zeta)=-2\left\langle\mathbf{x}_{\gamma}, N_{\zeta}\right\rangle\), where by \(\mathbf{x}_{c}\), for instance, we mean the vector with complex coordinates $$ \mathbf{x}_{p}=\left(\frac{\partial x}{\partial \zeta}, \frac{\partial y}{\partial \zeta}, \frac{\partial z}{\partial \zeta}\right) $$ c. Let \(f: U \subset C \rightarrow V \subset C\) be a one-to-one complex function given by \(f(u+i v)=x+i y=\eta\). Show that \((x, y)\) are isothermal parameters on \(S\) (i.e., \(\eta\) is a complex parameter on \(S\) ) if and only if \(f\) is analytic and \(f^{\prime}(\zeta) \neq 0, \zeta\) \& \(U\). Let \(\mathbf{y}=\mathbf{x} \circ f^{-1}\) be the correspond ing parametrization and define \(\phi(\eta)=-2\left\\{\mathbf{y}_{4}, N_{4}\right\\}\). Show that on \(x(U) \cap y(V) .\) $$ \phi(\zeta)=\psi(\eta)\left(\frac{3 \eta}{\partial \zeta}\right)^{2} $$ d. Let \(S^{2}\) be the unit sphere of \(R^{3}\). Use the stereographic projection (cf. Exercise 16, Sec. 2.2) from the poles \(N=(0,0,1)\) and \(S=(0,0,-1)\) to cover \(S^{2}\) by the coordinate neighborhoods of two (isothermal) complex parameters, \(\zeta\) and \(\eta\), with \(\zeta(S)=0\) and \(n(N)=0\), in such a way that in the intersection \(W\) of these coordinate neighborhoods (the sphere minus the two poles) \(\eta=\zeta^{-1}\). Assume that there exists on each coontinate neighborhood analytic functicns \(\psi(\zeta), \psi(n)\) such that \((+)\) holds in W. Use Liouville's theorem to prove that \(\varphi(\zeta)=0\) (hence, \(\psi(\eta)=0\) ). c. Let \(S \subset R^{3}\) be a regular surface with constant mean curvature homeomorphic to a sphere. Assume that there exists a conformal diffeomorphism \(\varphi: S \rightarrow S^{2}\) of \(S\) onto the unit sphere \(S^{2}\) (this is a consequence of the uniformization theorem for Riemann surfaces and will be assumed here). Let \(\bar{\zeta}\) and \(\bar{\eta}\) be the complex parameters corresponding ander \(\varphi\) to the parameters \(\zeta\) and \(\eta\) of \(S^{2}\) given in part \(\mathrm{d}\). By part a, the function \(\phi(\bar{\zeta})=((e-g) / 2)-\) if is analytic. The similar function \(\psi(\bar{p})\) is also 5.3. Complete Surfaces. Theorem of Hopf.Aioow \(93 t\) analytic, and by part c they are related by \((+)\). Use part d to show that \(\phi(\bar{\zeta})=0\) (hence, \(\psi(\bar{i})=0\) ). Conclude that \(S\) is made up of umbilical points and hence is a sphere. This proves Hopf's theorem.

Show that the point \(p=(0,0,0)\) of the paraboloid \(z=x^{2}+y^{2}\) has no conjugate point relative to a geodesic \(\gamma(s)\) with \(\gamma(0)=p\).
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