Submanifolds: Locally Deformable into Linear Patches

This lecture defines embedded submanifolds of linear spaces, using examples like a sphere, a cross, a cusp, and a double parabola. It explores the concept of smoothness and diffeomorphisms, connecting it to the ability to locally deform submanifolds into linear patches. The instructor introduces the inverse function theorem and proves that a subset is an embedded submanifold if and only if it can be locally deformed into linear spaces. The lecture delves into the construction of diffeomorphisms using local defining functions and the properties of their differentials, emphasizing the importance of invertibility. The main tool used is the inverse function theorem, which guarantees the existence of diffeomorphisms on suitable neighborhoods, ultimately leading to a comprehensive understanding of submanifolds.