Multivariable Calculus: Geodesic problem

[FluffyLlamas]

Homework Statement

Consider the parametrization of a torus:
$\tau(u,v)=((2+cos(v)cos(u),(2+cosv)sinu,sinv)$
The distance from the origin to the center of the tube of the torus is 2 and he radius of the tube is 1. Let the coordinates on $\mathbb R^3$ be $(x,y,z)$. If $p = \tau(u,v)$ then u is the angle between the x-axis and the line connecting the origin to the projection of p onto the xy-plane. A slice of the torus through the origin and p by a plane perpendicular to the xy-plane cuts out two circles on the torus, one of which contains p. The angle v is the angle from the center of the circle containing p out to p.

(a) Compute the first fundamental form matrix for $\tau$

(b) Find the lengths of (1,0) and (0,1) as elements of the tangent space at an arbitrary point (u,v). Explain the varying of these lengths by appealing to the picture above (textbook has a picture of a torus).

(c) What is the angle between (1,0) and (0,1) as elements in the tangent space at an arbitrary point (u,v)?

(d) Show that the circles $t \mapsto (c,t)$ where c is a constant are geodesics in [0,2∏] X [0,2∏] with the metric induced by $\tau$

(e) Show that the circles $t \mapsto (t,c)$ where c is a constant are not generally geodesics. When are they?

Homework Equations

I can't really think of any relevant equations, but the question itself is in this PDF:
http://people.reed.edu/~davidp/homepage/211.pdf

Problem is Chapter 6, #3 (Page 130)
Of course, throughout chapter 6, there are some definitions and such. I'll edit this post to include them if wanted.

Notes, from prof. on geodesics is here: http://people.reed.edu/~davidp/211.2012/lectures/41lecture.pdf

And this is a first semester multivariable calculus class for reference.

The Attempt at a Solution

So, I understand a-c, completed them, no trouble, I'll still write out my answers below (and edit with the work for them if necessary). But I missed the lecture on geodesics due to illness, and neither the textbook or notes that the professor puts up online aren't particularly helpful. I'm less worried about the specific problem and more about understanding geodesics themselves, so it's a more general question, I guess.

(a)
$\mathbb I = \begin{pmatrix} (cos(v)+2)^2 & 0\\ 0 & 1 \end{pmatrix}$

(b) $|(1,0)|_{(u,v)} = 10$
$|(0,1)|_{(u,v)} = (2+cos1)^2$

Haven't done the explanation part for this yet, not sure why it's the case. I'll have to think about it some more.

(c) $\theta = cos^{-1} ((3+7cos1)/(10(2+cos1)^2))≈ 1.4655 radians$

(d) This is where I'm lost. I've read through the notes and textbook many times, and I have no clue where to begin. Could anyone give me a tip on where to begin?

(e) Same as d

Thanks for any help.

 

[mighty2000]

I'll be sure to edit this post with my work as I work through it.Hello,

I would like to clarify some concepts before attempting to answer your questions. A geodesic is defined as the shortest path between two points on a curved surface. In the case of a torus, it can be thought of as the path that a string would take if it were stretched tightly around the surface.

Now, let's address your specific questions:

(d) In order to show that the circles t → (c,t) are geodesics, we need to show that they are the shortest path between any two points on the torus. This can be done by using the first fundamental form matrix to calculate the length of the curve and showing that it is minimized when t is constant. You can refer to your textbook or notes for the formula for calculating the length of a curve using the first fundamental form matrix.

(e) To show that the circles t → (t,c) are not generally geodesics, you can use the same approach as in part (d) and show that the length of the curve is not minimized when t is constant. In order for these circles to be geodesics, the torus would have to be a cylinder, where the shortest path between any two points is a straight line. You can also refer to your textbook or notes for more information on this.

In general, to understand geodesics better, I would recommend looking at some visualizations or animations of geodesics on different curved surfaces. This can help to develop an intuition for what they are and how they behave. Additionally, practicing with different examples and working through the calculations can also aid in understanding the concept.

I hope this helps and good luck with your studies!