Differential Geometry-Tubular surface

Question

I have an exam in a few days and I don't know how to solve this exercise from an old exam. Can you please help me?

Let $\alpha : \mathbb R -> \mathbb R^2 $ be the periodic parametrisation of a simply closed regular curve with positive curvature. Assume $\alpha$ is parametrised by arc length and let $n$ and $b$ be its normal and binormal vector respectively. Consider the tabular surface parametrised by $ f(s,t) = \alpha (s) + rn(s)cos(t) + rb(s)sin(t)$ with $ s,t \in \mathbb R , r>0$

1. Prove that such a tubular surface is a regular surface for r small enough
2. Compute the first and the second fundamental form
3. Compute its Gaussian curvature as a function of the curvature and torsion of $\alpha$
4. Show that the area of S is $2 \pi rl $
5. Compute the Euler characteristic of the surface using the Gauss-Bonnet theorem

I found that a surface is regular iff its Jacobi matrix has rang 2 or iff the gradient is $\neq 0$ So I used Frenet formula, i.e. $T(s)= \dot c(s),\; \dot T(s)=k(s)N(s),\; \dot N(s) = -T(s)k(s) + \tau(s)B(s),\; \dot B(s)=-\tau (s)N(s) \; $ where $T, N, B, k\; $and$ \;\tau $ are respectively the tangential vector, the normal vector, the binormal vector, the curvature and the torsion. Than I computed the gradient of f and I obtained: $f_s = T(s)(1-k(s)rcos(t))+r cos(t)\tau (s)b(s) - rsin(t)\tau(s)n(s)$ and $f_t = -rn(s)sin(t) + rb(s)cos(t)$. We can compare the two result and obtain $f_s = T(s)(1-k(s)rcos(t)) + \tau (s)f_t$. But I can't go on.

Fundamental forms

Sorry, I'm not able to create a matrix here so I did it in Latex and I did a screenshot.

For the second point I don't know how to simplify the computation, since my formula are very long.

For 3. I need 2. for compute the Gaussian curvature that is given by $\frac{det(II)}{det(I)}$

For 4. I use the formula $A(f)= \int_s \int_t \sqrt det(I)$ where $t\in [0,2\pi]$ and since the curve is parametrised by arc length $s\in [0,l]$ where $l$ is the length of the curve. But I'm not sure.

Than for 5. I have no Idea... there is a reasoning ora a formula to use?

Thanks

Answer 1

Here are some suggestions based on the work you've shown.

(1) The easiest thing to do is to compute the cross-product $f_s\times f_t$ and show that for $r$ small enough it will be everywhere nonzero. (Hint: $k(s)$ is bounded. Why?) You could also work with the formula you gave: Since $T(s)$ and $f_t$ are linearly independent, you can see directly that $f_s$ is parallel to $f_t$ precisely when the coefficient of $T(s)$ in your formula vanishes.

(2) Following through on the suggestion I made in (1), you will compute the cross-product and hence should be able to deduce the formula for the unit normal vector $N$. As should be suggested by drawing a picture, it should actually be the (unit) radius vector of the circle with center $\alpha(s)$, so you should get $\cos t\, n(s) + \sin t \,b(s)$ at the point $f(s,t)$.

You should now edit your post to include your (correctly simplified) first and second fundamental forms. They are not so messy, and you should be able to do this.

(4) This is correct.

(5) You need the Gauss-Bonnet Theorem. Presumably it was in your course, or this would not be an exam problem for you.

By the way, if you want some more reading with lots more examples and intuition, you might check out my differential geometry text. In particular, you can find Gauss-Bonnet clearly stated (and proved) in that.