/*! 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 4 Show that in a system of normal ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 4

Show that in a system of normal coordinates centered in \(p\), all the Christoffel symbols are zero at \(p\).

Short Answer

Expert verified
In normal coordinates centered at \(p\), the Christoffel symbols \(\Gamma^k_{ij}(p) = 0\).

Step by step solution

01

Understanding Normal Coordinates

Normal coordinates are a coordinate system where geodesics through a point can be represented as straight lines. At a point \(p\), the metric tensor is the identity matrix, and its first derivatives vanish.
02

Define Christoffel Symbols

The Christoffel symbols \(\Gamma^k_{ij}\) in terms of the metric tensor \(g_{ij}\) are given by: \[\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \frac{\partial g_{li}}{\partial x^j} + \frac{\partial g_{lj}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^l} \right)\]
03

Evaluate Metric at Point p

In normal coordinates centered at point \(p\), by definition, the metric tensor \(g_{ij}(p) = \delta_{ij}\) and \(\frac{\partial g_{ij}}{\partial x^k}(p) = 0\).
04

Substitute into Christoffel Symbols

Using the conditions from Step 3, substitute \(g_{ij} = \delta_{ij}\) and \(\frac{\partial g_{ij}}{\partial x^k} = 0\) into the Christoffel symbol equation. \[\Gamma^k_{ij}(p) = \frac{1}{2} g^{kl} \left( \frac{\partial g_{li}}{\partial x^j}(p) + \frac{\partial g_{lj}}{\partial x^i}(p) - \frac{\partial g_{ij}}{\partial x^l}(p) \right) = 0\]
05

Conclusion

Since all the partial derivatives of the metric tensor at \(p\) are zero, the Christoffel symbols \(\Gamma^k_{ij}(p)\) are also zero. This confirms that in a system of normal coordinates centered at \(p\), all Christoffel symbols vanish at \(p\).

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.

The Metric Tensor
The metric tensor is a fundamental concept in differential geometry. It provides a way to measure distances and angles between vectors in a manifold. Mathematically, the metric tensor, usually denoted as \( g_{ij} \), is a symmetric rank-2 tensor that varies smoothly from point to point on a manifold. In simpler terms, it defines the geometry of the space.

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

Compute the Christoffel symbols for an open set of the plane a. In Cartesian coordinates. b. In polar coordinates. Use the Gauss formula to compute \(K\) in both cases.

Justify why the surfaces below are not pairwise locally isometric: a. Sphere. b. Cylinder. c. Saddle \(z=x^{2}-y^{2}\).

Compute the Euler-Poincar茅 characteristic of a. An ellipsoid. b. The surface \(S=\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{10}+z^{6}=1\right\\}\).

Let \(\alpha: I \rightarrow R^{3}\) be a regular parametrized curve with \(k(t) \neq 0, t \in I\). Let \(\mathbf{x}(t, v)\) be its tangent surface. Prove that, for each \(\left(t_{0}, v_{0}\right) \in I \times\) \((R-\\{0\\})\), there exists a neighborhood \(V\) of \(\left(t_{0}, v_{0}\right)\) such that \(\mathbf{x}(V)\) is isometric to an open set of the plane (thus, tangent surfaces are locally isometric to planes).

Let \(T\) be a torus of revolution. Describe the image of the Gauss map of \(T\) and show, without using the Gauss-Bonnet theorem, that $$ \iint_{T} K d \sigma=0 $$ Compute the Euler-Poincar茅 characteristic of \(T\) and check the above result with the Gauss-Bonnet theorem.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.