if $n>2$ and the $C^{1}$-curve $\gamma = (\gamma_1, ... , \gamma_n)$, lies on the sphere $\mid x\mid=1$, (an object of dimension $n-1>1$) then the $(n-1)-\dim$ volume (area if $n=3$) of its image on the sphere is zero. (Take $n=3$ to fix ideas, and explain why the statement is wrong when $n=2!$) Hence the image of gamma can not have interior points (relative to the sphere); in particular, it would be far from a Peano curve!!
How to prove the last statement? Hint: On a small portion a sphere is practically flat.
This question is the next part of this question Image of continuously differentiable curve $\gamma:[0,1]\rightarrow \mathbb{R}^n$ has volume zero Please if someone can give me an idea to do this. I will be thankful for it.
At each point $p$ on your curve choose a small neighbourhood $B_p$ (on your sphere), such that $B_p$ is contained in a coordinate chart. Now the coordinate version of your curve is inside $B_p$, is a curve $\tilde \gamma_p $ in $\mathbb R^{n-1}$. Since $n>2$ you get (from the question your referenced to) that the $n-1$-volume of $\tilde \gamma_p$ is 0. Since this is the case for every point $p$ on your curve the entire volume is 0.