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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 \(p\).

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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.

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Most popular questions from this chapter

Let \(S \subset R^{3}\) be a regular, compact, connected, orientable surface which is not homeomorphic to a sphere. Prove that there are points on \(S\) where the Gaussian curvature is positive, negative, and zero.

Prove that an orientable compact surface \(S \subset R^{3}\) has a differentiable vector field without singular points if and only if \(S\) is homeomorphic to a torus.

Consider the torus of revolution generated by rotating the circle $$ (x-a)^{2}+z^{2}=r^{2}, y=0, $$ about the \(z\) axis \((a>r>0)\). The parallels generated by the points \((a+r, 0),(a-r, 0),(a, r)\) are called the maximum parallel, the minimum parallel, and the upper parallel, respectively. Check which of these parallels is a. A geodesic. b. An asymptotic curve. c. A line of curvature.

a. Show that if a curve \(C \subset S\) is both a line of curvature and a geodesic, then \(C\) is a plane curve. b. Show that if a (nonrectilinear) geodesic is a plane curve, then it is a line of curvature. c. Give an example of a line of curvature which is a plane curve and not a geodesic.

Surfaces of Liouville are those surfaces for which it is possible to obtain a system of local coordinates \(\mathbf{x}(u, v)\) such that the coefficients of the first fundamental form are written in the form $$ E=G=U+V, \quad F=0 $$ where \(U=U(u)\) is a function of \(u\) alone and \(V=V(v)\) is a function of \(v\) alone. Observe that the surfaces of Liouville generalize the surfaces of revolution and prove that (cf. Example 5) a. The geodesics of a surface of Liouville may be obtained by integration in the form $$ \int \frac{d u}{\sqrt{U-c}}=\pm \int \frac{d v}{\sqrt{V+c}}+c_{1} $$ where \(c\) and \(c_{1}\) are constants that depend on the initial conditions. b. If \(\theta, 0 \leq \theta \leq \pi / 2\), is the angle which a geodesic makes with the curve \(v=\) const., then $$ U \sin ^{2} \theta-V \cos ^{2} \theta=\mathrm{const} $$ (Notice that this is the analogue of Clairaut's relation for the surfaces of Liouville.)
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