Embedded submanifolds

Question

Set $L= \left\{ (x,y)\in \mathbb{R} : x^3=y^5\right\} $. Consider the parametrization of the curve $ t \to \left(t^5,t^3\right) $. Then the derivate of this curve at $0$ is $0$. Hence $L$ is not an embedded submanifold?

Thanks.

Answer 1

Having one bad parametrization does not necessarily prove that there can't be a good one. But you should know something an embedded submanifolds that doesn't depend on parametrization. They have tangent spaces at each point. Does this? Prove your answer. Then it must be the case that locally the submanifold is a graph (of a smooth function) over the tangent plane. Can you prove this?

EDIT: One of my favorite results (which has appeared on MathStackExchange before) is this: If $M$ is a $k$-dimensional embedded submanifold of $\Bbb R^n$, for every $p\in M$, $M$ can be written in some neighborhood of $p$ as the graph of a smooth function defined on one of the standard $k$-dimensional coordinate planes. (You can prove this by proving that the tangent plane of $M$ must project isomorphically onto at least one of those standard $k$-dimensional planes.)