2

I'm studying calculus and our professor gave us two definitions of manifolds:

$1) M\subset{R^n}$ is a k-dimensional manifold if for every $x\in M$ there exists a neighborhood $U_x\subset R^n$ such that $M\cap U_x$ is a graph, i.e there exist indexes $i_1,...,i_k$ and $j_1,...j_{n-k}$ which their union is $\{1...n\}$ and an open subset $V\subset R^k$ and a smooth function $f:V\to R^{n-k}$ such that $M\cap U_x=\{x\in R^n | (x_{j_1},...,x_{j_{n-k}})=f(x_{i_1},...x_{i_k}), (x_{i_1},...x_{i_k})\in V\}$.

$2)M\subset{R^n}$ is a k-dimensional manifold if for every $x\in M$ there exist a neighborhood $U_x$ and a diffeomorphism $\phi:U\to \phi(U)$ such that $\phi(M\cap U)=\phi(U)\cap (R^{n-k}\times\{0\}_{n-k})$ when $\{0\}_{n-k}$ represents the n-k dimensional $0$-vector.

The first thing I'm having trouble is understanding the meaning of the second definition. I miss some intuition to why it is defined this way. The second thing is understanding why the two definitions are equivalent.

Tamir Shalev
  • 321
  • 1
    I think you have to understand that both definitions are definitions of submanifolds for $R^n$. In general, a manifold does not need to be subset of $R^n$ (although we can always find embedding to $R^{2k}$ due to the Whitney's embedding theorem – Li Chun Min Mar 07 '19 at 00:21
  • And the second definition follows from the third definition that “$M$ is a smooth submanifold of $N$ if $M$ is a smooth manifold and the inclusion map is an immersion” due to the immersion theorem (sorry for bringing in extra definitions that confuse you xD – Li Chun Min Mar 07 '19 at 00:25
  • 1
    Anyway, (I am also learning about manifolds) I think the second def essentially means $M$ locally looks like a $k$-dimensional plane on the Euclidean space and the equivalent of the two defs can established using the implicit function theorem?...(probably someone is writing an answer already – Li Chun Min Mar 07 '19 at 00:51
  • https://math.stackexchange.com/questions/2900577/regular-submanifold-in-mathbbr2-is-locally-a-graph – Moishe Kohan Mar 07 '19 at 23:59

0 Answers0