3

I have a final coming up in Differential Geometry and we got a review worksheet and I am having serious trouble with two problems. I'm still chugging along at them but I need help understanding. I know we learned the 2nd problem (#5) while I was sick last week, so I'm deep in my book trying to comprehend what i missed. Any help would be appreciated!

Problem 1

enter image description here

I have been working on 5 and I have come up with the following, which makes it seem way easier than it should be so I'm probably wrong

plug in t=0 which leads to both being (0,0). I end up using the equation cosx=F/sqrt(EG) and plugging in the values for the first fundamental form i get pi/2, which i remember reading if F=0 then all curves are orthogonal at their intersections. Am i right about any of this?

I'm stuck on 2, but i Haven't put much effort into it yet compared to 5.

Metzky
  • 41
  • What is your definition of a regular curve? – Olivier Bégassat Dec 13 '13 at 00:41
  • Essentially a regular curve's derivative is larger than zero, so the curve never slows down to a stop – Metzky Dec 13 '13 at 00:47
  • If the only condition you place on your curve is that its first derivative never vanishes, then the equivalence in the first question is false. Sometimes a regular curve (in $\mathbb{R}^3$) is defined as a $C^3$ curve whose first, second and third derivative are everywhere linearly independent. I think with this definition the equivalence ought to be true. In fact, the linear independence at every point of the first and second derivative should be enough. – Olivier Bégassat Dec 13 '13 at 00:56
  • I'm sorry I'm lost. So is the question wrong then? – Metzky Dec 13 '13 at 01:10
  • Either it's wrong or the definition of regular curve you gave me is not the one the author uses. I think the question would be right if you defined a regular curve as a curve such that for all $x\in (a,b)$, the family $(\alpha'(x),\alpha''(x))$ is free. Although that's not quite standard, it is more common to ask that for all $x\in (a,b)$, the family $(\alpha'(x),\alpha''(x),\alpha'''(x))$ is a basis. – Olivier Bégassat Dec 13 '13 at 01:10
  • i found the section in my book "a parametrized differentiable curve a:I -> R^3 is said to be regular if a'(t) =/ 0 for all t in I" – Metzky Dec 13 '13 at 01:15
  • Well then I can describe a counter example to the first question. – Olivier Bégassat Dec 13 '13 at 01:16
  • so there is no way to answer this question? What is a counter example? – Metzky Dec 13 '13 at 01:18
  • if it were a plane curve isn't it two dimensional meaning one of the components must be zero? nvm that was stupid – Metzky Dec 13 '13 at 01:31

1 Answers1

-1

The counter example I have in mind is a curve you built in two steps. Consider a smooth curve $\alpha$ defined on $[0,5]$ that satisfies $$\alpha(t)=\begin{cases} (t,0,0) & \text{for }t\in[0,1] \\(2,1+t,0) & \text{for }t\in[2,3] \\(2,3,1+t) & \text{for }t\in[4,5] \end{cases}$$ and has $\alpha([0,3])\subset\lbrace z=0\rbrace$ and $\alpha([2,5])\subset\lbrace x=2\rbrace$. Smooth curves satisfying these constraints can be constructed, one could even give explicit formulas using integrals of smooth bump functions for the parts of the curve that I have not described.

Such a regular curve (in your sense) will satisfy $(\alpha'\times\alpha'')\cdot\alpha'''\equiv0$ yet is manifestly not planar. The reason that quantity is constant equal to $0$ is that for every point $x\in[0,5]$, there is an open neighborhood $V$ of $x$ with $\alpha(V)\subset\mathcal{P}$ where $\mathcal{P}$ is one of the planes $\lbrace z=0\rbrace$ and $\lbrace x=2\rbrace$. The problem should disappear if we impose that at every point $x$ of the domain of $\alpha$, the first and second derivative $\alpha'(x)$ and $\alpha''(x)$ be linearly independent.

Olivier Bégassat
  • 20,248
  • Ok i get what you are saying. There are regular curves that satisfy the problem without being planar? – Metzky Dec 13 '13 at 01:41
  • Yes. But as I said, I think the problem is with the definition of regular curve. – Olivier Bégassat Dec 13 '13 at 01:42
  • i see where the problem lies. I'm going to look in the book some more and see if i can't find something else to help clear up the confusion. Thanks for all the help! – Metzky Dec 13 '13 at 01:45
  • The unexplained downvotes are always a treat (related recent meta post: http://meta.math.stackexchange.com/questions/11992/up-and-downvote-statistics) – Olivier Bégassat Dec 17 '13 at 03:33