I am reading Linear algebra and analysis by André Lichnerowicz and there is a notion on which I find nothing on the internet.
First the context : We consider $E$ and $F$ real Euclidean spaces of dimension m and n ($m\leq n$).We denote by $E_{z}$ the space of vectors with origin $z\in E$ that is the vector space
$$ \{ y\in E : y = x - z, x\in E\} $$
The differential of a function $f$ at a point x is noted $L(x,dx)$.
Now for a differentiable function $f$ and a fixed $x$ we assign to every vector $dx\in E_{x}$ the corresponding vector $dy = L(x,dx)$ of $F_{y}$ and we represent by $G_y$ the set of vectors $dy$ thus obtained.
We consider functions $f$ from a domain $\Delta\subset E$ to $\Delta_1\subset F$ that are differentiable and satisfy the followings conditions :
$f$ is bijective
For all $x\in \Delta$, the differential $L(x,dx)$ from $E_{x}$ onto $G_{y}$ is bijective
We call these functions the functions of class $H_1$.
Now the definition of a piece of a continuously differentiable manifold. Here the domain $\Delta$ is an m dimensional block of the space $E$.
The domain $\Delta_1$ which is the image of the $m-block$ $\Delta$ under a function of class $H_1$ is called a piece of a continuously differentiable manifold of dimension $m$ embedded in F.
In order to have a better understanding of this notion I tried to see other sources on the internet but I cannot find this notion elsewhere. Have you ever heard about this notion of piece of continuously differentiable manifold ?
Thank you !