im studyin the book "Introduction to differential topology" by Th. Bröcker and K. Jänich and I'm stuck with two of the exercices in Chapter 9 on isotopy.
A short overview (for those who are familiar with the topic: skip until the fat word): Let $N$ and $M$ be two smooth manifolds and let $h_0:N\rightarrow M$ and $h_1:N\rightarrow M$ be two embeddings of $N$ in $M$. The embeddings are said to be isotopic if there exists $h:[0,1]\times N\rightarrow M$ differentiable such that $$h(t,\cdot):N\rightarrow M, p\mapsto h(t,p)$$ is an embedding for every t and $h(0,\cdot)=h_0$, $h(1,\cdot)=h_1$.
$h_0$ and $h_1$ are said to be diffeotopic if there exists $d:[0,1]\times M\rightarrow M$ differentiable such that $d(t,\cdot): M\rightarrow M$ is a diffeomorphism for every t (we call $d$ a diffeotopy) and $f(t,\cdot):=d(t,\cdot)\circ h_0$ is an isotopy between $h_0$ and $h_1$.
Now the Question: The first exercice is to prove, that two embeddings $h_0, h_1: \mathbb{R}\rightarrow\mathbb{R}$ that keep the orientation of $\mathbb{R}$ are isotopic. The second is to find two embeddings $h_0, h_1: \mathbb{R}\rightarrow\mathbb{R}$ that keep the orientation of $\mathbb{R}$ but are not diffeotopic.
While the first one seems quite obvious to me i still don't really know how to prove it. The second statement seems quite unintuitive to me and I don't really have an Idea how to tackle the problem. Do you have any help?