I have the following definition of the submanifold as you can see in the image, I am trying to construct an example to understand the condition that a submanifold is satisfying. But I think I am making some mistake, I will be grateful if someone helps me out,
So let the manifold $S=\mathbb{R^2}$ of dimension 2 and the submanifold $M=\mathbb{R}$ of dimension 1 as defined in the image let $\xi^1,\xi^2$ and $u^1=u$ be the coordinates of the manifold $S$ and submanifold $M$ respectively.Now I wanted to look for the condition $(i),(ii),(iii)$
as condition $(i)$ says that the restriction of coordinate function $\xi^i|M$ to be $C^{\infty}$,as we know that $\xi^i:S\to \mathbb{R}$ is defined as $\xi^i(\xi^1,\xi^2)=\xi^1 \text{ if i=1 and }=\xi^2 \text{ if i=2 }$ so if I restrict it to $M=\mathbb{R}$ then we will have $\xi^i|M:M\to \mathbb{R}$ such that $\xi^i(u)=u$ since in $M$ we have only $u$ as a coordinate and this is $C^{\infty}$ so the condition (i) is verified.
for condition (ii) we have that $B_{a}^{i}=(\frac{\partial\xi^i|M}{\partial u^a})_p$ and $B_a=[B_{a}^{1},....,B_{a}^{n}]\in \mathbb{R^n}$ then for each point $p\in M$ the set $\{B_1,B_2,....,B_m\}$ are linearly independent.
so if we convert this condition according to our case we will have to check that the set $\{B_1\}$ is linearly independent which means it should not be zero so we have $B_1=[b_{1}^{1}]$ and $B_{1}^{1}=(\frac{\partial \xi^1|M}{\partial u})=1$
I wanted to know that is my construction is correct?? If yes then can someone explain what this 2nd condition is trying to say?