Let ($\mathbb{R}^n$,$|.|_p$) the normed space
with $\left | x \right |_{p}=(|x_1|^p+\cdots+|x_n|^p)^{1/p}$ the p-norm $p>1$
let $\gamma:[a,b] \rightarrow \mathbb{R}^n $ be a rectifiable curve parametrised by arc lenght
it's clear that a lot of thnigs change from the natural euclidean geometry (unit disk , the lenght ...)
my Question Question are : 1_ how we define that two vectors $x,y \in \mathbb{R}^n $ are orthogonal in this case ?
2_how can we define the curvature in this case ($k(t)=| \gamma^{}"(t)| _{p}$) ?
3_ in general what changes compared to euclidean geometry ? did you know book or a lecture notes on differential Geometry treat this case of non euclidean geometry ?