A diffeomorphism is a smooth, bijective function between two differentiable manifolds that has a smooth inverse. It essentially means that the two manifolds are "diffeomorphic" or "smoothly equivalent" to each other.
An embedded submanifold is a subset of a larger manifold that is itself a manifold. This means that it has a well-defined notion of smoothness and can be described by coordinates in a neighborhood.
Diffeomorphisms are often used to transform or map one manifold onto another, and this can be used to show that two manifolds are diffeomorphic. Embedded submanifolds are often defined using diffeomorphisms, such as by embedding a smaller manifold into a larger one.
Diffeomorphisms and embedded submanifolds are important concepts in differential geometry and have many applications in physics, such as in general relativity and fluid dynamics. They are also used in computer graphics and computer vision to model and analyze 3D objects.
One challenge with using diffeomorphisms and embedded submanifolds is that they can be difficult to visualize and manipulate, especially in higher dimensions. Additionally, finding a diffeomorphism between two manifolds can be a complex and computationally intensive task.