Manifold and hyperplane

Question

Can someone explain me the relation between manifold and hyperplane. I saw a definition but I am not able to connect the idea. The definition is

A set Γ ⊂ $R^n $ is called a k–dimensional $C_m$manifold (k < n), if for every point x ∈ Γ there exists a neighborhood $U_x$ such that Γ ∩ $U_x$ can be be written as the graph of a m–times differentiable function from $R^k$ into $R^{n-k}$. It is clear, what we mean by calling a manifold analytic. Typically, (n − 1)– dimensional manifolds are called hyperplanes.

My doubt is if the mapping is from $R^k to R^{n-k} $ won‘ t the codomain become $R^{n-k} $ then how does it suffices the explanation for hyperplane.

P.S

This is the first time I am coming across manifold. So I know only basic definitions.

Answer 1

It is essentially enough to think of Hyperplanes in $\mathbb{R}^3$. In that they are translation of a 2-dimensional subspace of $\mathbb{R}^2$, in other words a plane. Now, if we replace $\mathbb{R}^3$ with $\mathbb{R}^n$, then naturally we will take n-1 dimensional subspace and translate it by a vector.

Manifolds come in the picture when you are doing things locally, that is on arbitrarily small neighborhoods. In such a setup, it is enough to assume that the space you're dealing with, is like $\mathbb{R}^m$ in a small neighborhood around each point (Often called "locally" like $\mathbb{R}^3$).

Now Hyperplanes are a very special case of manifolds. They are locally like $\mathbb{R}^{(n-1)}$

Answer 2

Well perhaps firstly one could define an affine manifold, and the related concept of a hyperplane. Lets consider two and three dimensions.

A subset $S$ of a linear space $V$ is an affine manifold of $V$ if $S = Z + x$ for some linear subspace $Z$ of $V$ and some $x \in V$.

So, sticking to $\mathbb{R}^2$, what do manifolds and hyperplanes look like ? Well, firstly, remember that there are two classes of linear subspace in R2 : 0, and any line through the origin. Thus, we have two classes 22 of affine manifold: any point, and any line. The latter is clearly the class of hyperplanes in $\mathbb{R}^2$. Similarly, in $\mathbb{R}^3$ we have three classes of affine manifolds: points, lines and planes, with planes being the hyperplanes.

This means that in $\mathbb{R}^2$, any hyperplane can be defined by a function of the form $n_1x_1+n_2x_2 = b$, (as any line can be written in this form). In other words, they are described by the set

$$\{x|<n \cdot x>=b\}$$

for some $n \in \mathbb{R}^2$ and $b \in \mathbb{R}$. In fact, it turns out that this is a general characterization of hyperplanes in $\mathbb{R}^n$. Then we consider a hyperplane

A set $S \in \mathbb{R}^n$ is a hyperplane if and only if $\{x \in \mathbb{R}^n|<n \cdot x>=b\}$ for some $n \in \mathbb{R}^n / \emptyset, b \in \mathbb{R} $