Compute the lengths of $\gamma,\sigma$ and $\eta$ with respect to $g$.

Question

Consider r the manifold $M = \{(x^1,x^2) \in \mathbb{R}^2 | x^2 > 0\}$ equipped with the Riemannian metric

$$g(x)=\frac{1}{(x^2)}^2 (dx^1 \otimes dx^1+dx^2 \otimes dx^2)$$

a. Sketch the following three curves in $M$:

1. $\gamma (t)=(1-2t,1), \ t\in [0,1]$

2. $\sigma (t)=(\sqrt{2} \cos(t),\sqrt{2} \sin(t)), \ t\in [\frac{\pi}{4},\frac{3\pi}{4}]$

3. $\eta (t)= \begin{cases} (1,1+t) & t\in [0,1]\\ (1-2(t-1),2) & t \in [1,2] \\ (-1,2-(t-2)) & t\in [2,3] \end{cases} $

Here is what I got

1

b. Compute the lengths of $\gamma,\sigma$ and $\eta$ with respect to $g$. Which curve is the shortest? Which is the longest?

I am not sure how to calculate the length in manifolds. Can anyone calculate the length of one of the curves and I will do the others. I know that I have to use the formula

$$\mathcal{L}_g (\gamma)=\int_0^1 ||\dot \gamma(t)||_g dt$$

but I am not sure how to do this with respect to $g$, especially for $\eta$ since it contains $3$ curves.

Any help would be appreciated. Thanks.

Answer 1

Write $g|_p$ for the metric evaluated at the point $p = (p^1,p^2) \in M$, so that $$g|_p =\frac 1 {[p^2]^2} \left(dx^1|_p \otimes dx^1|_p + dx^2|_p \otimes dx^2|_p\right).$$ (Here $dx^j|_p$ is the standard differential form in the $j$-th direction on $M$, evaluated at $p$.) For any curve $\alpha : [a,b] \to M$ you should calculate $$\begin{split} \mathcal L_g (\alpha) &= \int_a^b \sqrt{g|_{\alpha(t)}(\dot\alpha(t),\dot\alpha(t))} dt =\int_a^b \sqrt{g_{ij}({\alpha(t)})\ \dot\alpha^i(t)\ \dot\alpha^j(t)} dt \\ &=\int_a^b \sqrt{\frac{1}{[\alpha^2(t)]^2}\left([\dot\alpha^1(t)]^2 + [\dot\alpha^2(t)]^2\right)} dt. \end{split}$$ Also keep in mind that integrals are additive, so if $\alpha$ is defined piecewise, you can always split the integral up into those pieces.

For a specific calculation, see here.