0
$\begingroup$

A subset $M \in \mathbb{R}^n$ is called a smooth k-dimensional submanifold of $\mathbb{R}^n$, $k \leq n$, if any point $x \in M$ has a neighborhood $O_x$ in $\mathbb{R}^n$ in which $M$ is described in one of the following ways:

  • (1) there exists a smooth vector-function $$F:O_x \to \mathbb{R}^{n-k}, \text{ } rank \frac{dF}{dx}|_q=n-k $$ such that $O_x \cap M = F^{-1}(0)$

  • (2) there exists a smooth vector-function $$f:V_0 \to \mathbb{R}^{n}$$ from a neighborhood of the origin $0 \in V_0 \subset \mathbb{R}^k$ with $$f(0)= x \text{ and } rank \frac{df}{dx}|_0=k $$ such that $f: V_0 \to O_x \cap M$ is homeomorphism.

  • (3) there exists a smooth vector-function $$\Phi:O_x \to O_0 \subset \mathbb{R}^{n},$$ onto a neighborhood of the origin $0 \in O_0 \subset \mathbb{R}^n$ with $$\text{ } rank \frac{d \Phi}{dx}|_x=n $$ such that $ \Phi(O_x \cap M)= \mathbb{R}^k \cap O_0$

I am trying to show the equivalence of these definitions and my first question is why is each definition imposing itself on the rank?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.