In this lecture we prove the manifold version of the Implicit Function Theorem. Along the way we introduce the concept of a submanifold $M$ of a larger manifold $N$. These come in two flavours: immersed submanifolds and embedded submanifolds. Our treatment of submanifolds allows us to finally make good on the promise from Lecture 2 and recover the “intuitive” definition of the tangent space of a manifold.

Next lecture we will show that every manifold $M$ is diffeomorphic to an embedded submanifold of some Euclidean space $\mathbb{R}^n$ — this is known as the Whitney Embedding Theorem.