/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 A geodesic \(\gamma:[0, \infty) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 6

A geodesic \(\gamma:[0, \infty) \rightarrow S\) on a surface \(S\) is a ray issuing from \(\gamma(0)\) if it realizes the (intrinsic) distance between \(\gamma(0)\) and \(\gamma(s)\) for all \(s \in[0, \infty)\). Let \(p\) be a point on a complete, noncompact surface \(S\). Prove that \(S\) contains a ray issuing from \(p\).

Short Answer

Expert verified
S contains a geodesic ray issuing from any point \(p\) due to its completeness and noncompactness.

Step by step solution

01

- Understand the Problem

Identify the key concepts in the question: a geodesic \(\gamma:[0, \infty) \rightarrow S\) on a surface \(S\) is a ray if it realizes the distance between \(\gamma(0)\) and \(\gamma(s)\) for all \(s \in[0, \infty)\). The goal is to prove that a complete, noncompact surface \(S\) contains a ray issuing from any point \(p\) on it.
02

- Choose a Starting Point and Create a Sequence

Choose a point \(p \in S\). Since \(S\) is noncompact, it implies that for a given point \(p\), there exist points arbitrarily far from \(p\) on \(S\). Therefore, create a sequence of points \((p_n)\) on \(S\) such that the intrinsic distance \(d(p, p_n) \rightarrow \infty\) as \(n \rightarrow \infty\).
03

- Use Completeness to Obtain Convergent Subsequences

Since the surface \(S\) is complete, every Cauchy sequence has a convergent subsequence. Consider the points \(p_n\) and ensure that each pair \((p, p_n)\) forms a Cauchy sequence. Consequently, every convergent subsequence must approach the desired intrinsic distance.
04

- Construct a Geodesic

Construct a geodesic \((\gamma)\) that connects \(p\) to each point \(p_n\). By definition of completeness and noncompactness, there will always be a path (geodesic) connecting \((p, p_n)\) that satisfies the distance requirement.
05

- Extend the Geodesic to Infinity

Using the property that \((S, d)\) is a geodesically complete space, extend the geodesic \(\gamma(s)\) to an infinite geodesic path that issues from \(p\) and realizes the intrinsic distance between \(\gamma(0) = p\) and \(\gamma(s)\) for all \(s \in[0, \infty).\)
06

- Conclusion

Conclude that \(\gamma(s)\) is a geodesic ray issuing from \(p\), for all \(s \in[0, \infty).\) Therefore, the complete, noncompact surface \(S\) contains a ray issuing from any point \(p\) on it.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Geodesics
In mathematics, a geodesic is the shortest path between two points on a curved surface. Think of it as the straight line of a curved space. For example, on a sphere like Earth, the shortest path between two points is an arc of a great circle鈥攖his is a geodesic. Geodesics are important because they generalize the notion of a straight line to curved surfaces and can be used in different fields such as physics, geography, and computer graphics. In the context of the given problem, a geodesic ray is a specific type of geodesic that extends infinitely from a starting point while maintaining the shortest distance property.
Noncompact Surfaces
A noncompact surface is a type of surface that extends infinitely in at least one direction. Picture an endless plane or an infinitely long cylinder. These surfaces don't 'close in' on themselves. This is in contrast with compact surfaces, which are finite and have a closed boundary, like a sphere or a torus. Noncompact surfaces are vital when studying geodesics because they ensure the possibility of infinitely extending paths. In our problem, the noncompact nature of the surface allows for points to be arbitrarily far from any starting point. Consequently, it is possible to form sequences and extend geodesics indefinitely.
Complete Surfaces
A surface is said to be complete if every Cauchy sequence of points on the surface converges to a point that also lies on the surface. In simpler terms, you can think of it as a surface where geodesics can be extended infinitely without hitting any 'edge' or lacking points. This makes navigation and calculation on the surface predictable and well-behaved. In our problem, the completeness of the surface allows us to ensure that any sequence of points moving infinitely away from a starting point will have a well-defined path, aiding in constructing a geodesic ray.
Intrinsic Distance
The intrinsic distance on a surface is the distance measured along paths within the surface itself, not through the surrounding space. It's like measuring distances on a map, rather than as-the-crow-flies distances in the air above the map. This is particularly crucial when dealing with curved surfaces because the shortest distance is not always a straight line if viewed from outside the surface. In the given exercise, the intrinsic distance ensures that the geodesics considered are truly the shortest paths within the surface, allowing for a geodesic ray to correctly represent an infinite extension from a point while maintaining minimal distance at each step.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(Calculus of Variations.) Geodesics are particular cases of solutions to variational problems. In this exercise, we shall discuss some points of a simple, although quite representative, variational problem. In the next exercise we shall make some applications of the ideas presented here. Let \(y=y(x), x \in\left[x_{1}, x_{2}\right]\) be a differentiable curve in the \(x y\) plane and let a variation of \(y\) be given by a differentiable map \(y=y(x, t)\), \(t \in(-\epsilon, \epsilon)\). Here \(y(x, 0)=y(x)\) for all \(x \in\left[x_{1}, x_{2}\right]\), and \(y\left(x_{1}, t\right)=\) \(y\left(x_{1}\right), y\left(x_{2}, t\right)=y\left(x_{2}\right)\) for all \(t \in(-\epsilon, \epsilon)\) (i.e., the end points of the variation are fixed). Consider the integral 4\. First and Second Variations of Are Length, Bonnet's Theorem $$ I(t)=\int_{x_{1}}^{x_{2}} F\left(x, y(x, t), y^{\prime}(x, t)\right) d x, \quad t \in(-\epsilon, \epsilon) $$ where \(F\left(x, y, y^{\prime}\right)\) is a differentiable function of three variables and \(y^{\prime}=\partial y / \partial x\). The problem of finding the critical points of \(I(t)\) is called a variational problem with integrand \(F\). a. Assume that the curve \(y=y(x)\) is a critical point of \(I(t)\) (i.e., \(d I / d t=0\) for \(t=0\) ). Use integration by parts to conclude that \((I=d I / d t)\) $$ \begin{aligned} I(t) &=\int_{x_{1}}^{x_{2}}\left(F_{y} \frac{\partial y}{\partial t}+F_{y} \frac{\partial y^{\prime}}{\partial t}\right) d x \\ &=\left[\frac{\partial y}{\partial t} F_{y}\right]_{x_{1}}^{x_{2}}+\int_{x_{1}}^{x_{2}} \frac{\partial y}{\partial t}\left(F_{y}-\frac{d}{d x} F_{y^{\prime}}\right) d x \end{aligned} $$

(Kazdan-Wamer's Results.) a. Let a metric on \(R^{2}\) be given by $$ E(x, y)=1, \quad F(x, y)=0, \quad G(x, y)>0, \quad(x, y) \in R^{2} . $$ Show that the curvature of this metric is given by $$ \frac{\partial^{2}(\sqrt{G})}{\partial x^{2}}+K(x, y) \sqrt{G}=0 . $$ b. Conversely, given a function \(K(x, y)\) on \(R^{2}\), regard \(y\) as a parameter and let \(\sqrt{G}\) be the solution of \((*)\) with the initial conditions $$ \sqrt{G}\left(x_{0}, y\right)=1, \quad \frac{\partial \sqrt{G}}{\partial x}\left(x_{0}, y\right)=0 . $$ Prove that \(G\) is positive in a neighborhood of \(\left(x_{0}, y\right)\) and thus defines a metric in this neighborhood. This shows that every differentiable function is locally the curvature of some (abstract) metric. *. Assume that \(K(x, y) \leq 0\) for all \((x, y) \in R^{2}\). Show that the solution of part \(b\) satisfies \(\sqrt{G(x, y)} \geq \sqrt{G\left(x_{0}, y\right)}=1 \quad\) for all \(x\) Thus, \(G(x, y)\) defines a metric on all of \(R^{2}\). Prove also that this metric is complete. This shows that any nonpositive differentiable function on \(\mathrm{R}^{2}\) is the curvature of some complete metric on \(\mathrm{R}^{2}\). If we do not insist on the metric being complete, the result is true for any differentiable function \(K\) on \(R^{2}\). Compare J. Kazdan and F. Warner, "Curvature Functions for Open 2-Manifolds," Ann. of Math. 99 (1974), 203-219, where it is also proved that the condition on \(K\) given in Exercise 2 of Sec. \(5-4\) is necessary and sufficient for the metric to be complete.

Let \(\gamma:[0, l] \rightarrow S\) be a geodesic on a complete surface \(S\), and assume that \(\gamma(l)\) is not conjugate to \(\gamma(0)\). Let \(w_{0} \in T_{\gamma(0)}(S)\) and \(w_{1} \in T_{\gamma(l)}(S)\). Prove that there exists a unique Jacobi field \(J(s)\) along \(\gamma\) with \(J(0)=w_{0}\), \(J(l)=w_{1} .\)

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.

(Kneser Criterion for Conjugate Points.) Let \(S\) be a complete surface and let \(\gamma:[0, \infty) \rightarrow S\) be a geodesic on \(S\) with \(\gamma(0)=p\). Let \(K(s)\) be the Gaussian curvature of \(S\) along \(\gamma\). Assume that Covering Spaces; The Theorems of Hadamard $$ \int_{t}^{\infty} K(s) d s \leq \frac{1}{4(t+1)} \text { for all } t \geq 0 $$ ( \(k\) ) in the sense that the integral converges and is bounded as indicated. a. Define $$ w(t)=\int_{t}^{\infty} K(s) d s+\frac{1}{4(t+1)}, \quad t \geq 0 $$ and show that \(w^{\prime}(t)+(w(t))^{2} \leq-K(t)\). b. Set, for \(t \geq 0, w^{\prime}(t)+(w(t))^{2}=-L(t)\) (so that \(\left.L(t) \geq K(t)\right)\) and define $$ v(t)=\exp \left(\int_{0}^{t} w(s) d s\right), \quad t \geq 0 $$ Show that \(v^{\prime \prime}(t)+L(t) v(t)=0, v(0)=1, v^{\prime}(0)=0\). c. Notice that \(v(t)>0\) and use the Sturm oscillation theorem (Exercise 6) to show that there is no Jacobi field \(J(s)\) along \(\gamma(s)\) with \(J(0)=0\) and \(J\left(s_{0}\right)=0, s_{0} \in(0, \infty)\). Thus, if \((*)\) holds, there is no point conjugate to \(\mathrm{p}\) along \(\gamma\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.