Abstract :

The most common examples of smooth manifolds are seen to be subsets of the euclidean spaces, such as such as $\mathbb{S} ^n$, a subset of $\mathbb{R}^{n+1}$, is a manifold of dim $n$ on it's own right. It's then usual to ask which subsets of euclidean spaces are smooth manifolds on their own. One way to get satisfactory answer is to study various kind of smooth maps, namely immersions, embeddings, submersions, and examine their images and level sets. In this talk, we'll discuss some of the well known theorems, such as inverse function theorem & family, and their consequences on smooth manifolds and will see when does level sets or images of smooth maps defines smooth manifolds.