During my math studies, I got this question. I tried to solve it for a few days but didn't succeed.
Let $M$ be a $ k $-dimension surface in ${\mathbb{R}}^n$, $(1 \le k \lt n)$ and $x \in M$.
I need to prove that there is $\delta \gt 0$ and (at least) $ n - k $ different indexes $1 \le i_1 , i_2, \dots , i_{n-k} \leq n $ such that for all $ 0 \lt |t| \lt \delta $, $ x + t e_{i_j} \notin M $ (for all $1 \le j \le n - k$).
i.e. $\{e_1, \dots ,e_n\}$ is the stardard basis of ${\mathbb{R}}^n$.
Does anyone have an idea of how can I prove this? ($M$ is a smooth surface)