Diffeomorphism vs Local Diffeomorphism in Differential Geometry

Differential geometry is a fundamental branch of mathematics that studies the properties of curves, surfaces, and manifolds. Central to this field are concepts such as diffeomorphisms and local diffeomorphisms. These concepts are crucial for understanding the smooth structure of manifolds and the relationships between differentiable spaces. Here, we will delve into the definitions and differences between a diffeomorphism and a local diffeomorphism, providing both formal definitions and intuitive explanations for better comprehension.

What Is a Diffeomorphism?

A diffeomorphism is a smooth bijective function between two manifolds that has a smooth inverse. Formally, let M and N be two smooth manifolds. A map f: M to N is a diffeomorphism if it satisfies the following conditions:

The function f is smooth (infinitely differentiable). The function f is bijective (one-to-one and onto). The inverse function F^{-1}: N to M is also smooth.

Diffeomorphisms are essential in various areas of mathematics, particularly in the study of manifold topology and differential geometry, as they preserve the smooth structure and homeomorphic properties of manifolds.

Understanding Local Diffeomorphisms

A local diffeomorphism is a map that is smooth and behaves like a diffeomorphism in a neighborhood of every point in its domain. More specifically, a map f: M to N is a local diffeomorphism if it meets the following criteria:

The function f is smooth. For every point p in M, there exists a neighborhood U of p such that f_U: U to f(U) is a diffeomorphism onto its image.

Unlike a diffeomorphism, a local diffeomorphism does not need to be globally bijective. It can be many-to-one and does not necessarily cover the entire target manifold, but it retains local invertibility in small neighborhoods. This distinction is significant in understanding the non-global properties of smooth mappings between manifolds.

Key Differences Between Diffeomorphisms and Local Diffeomorphisms

The primary difference between a diffeomorphism and a local diffeomorphism lies in their global and local properties:

Diffeomorphism

_GLOBAL BIJECTIVITY_: A diffeomorphism is a one-to-one and onto function between manifolds. _SMOOTHNESS INVERSE_: The inverse of a diffeomorphism is also smooth. _COMPLETENESS_: A diffeomorphism ensures that the source and target manifolds are globally homeomorphic and their smooth structures are preserved.

Local Diffeomorphism

_LOCAL BIJECTIVITY_: A local diffeomorphism behaves like a diffeomorphism in a small neighborhood but may not be global. _NO REQUIREMENT FOR GLOBAL BICJECTIVITY_: A local diffeomorphism can be many-to-one or not cover the entire target manifold. _LOCAL INVERTIBILITY_: Each point in the domain has a neighborhood where the map is bijective and smooth.

The key takeaway is that while every diffeomorphism is a local diffeomorphism, not every local diffeomorphism is a diffeomorphism. This distinction is crucial in the study of manifolds, where global properties and local behaviors need careful consideration.

Examples and Intuitive Understanding

Visualizing these concepts can be helpful. Consider a flat square sheet of paper that is rolled into a tube:

The map from the original “sheet of paper” manifold to the new “tube” manifold is a local diffeomorphism but not a diffeomorphism. At any point on the sheet of paper, you can "cut out" a circle that maps 1:1 onto a part of the tube with no overlap. However, when you consider the sheet of paper as a whole, it overlaps itself when mapped onto the tube.

This property of overlapping nearby but not overall is precisely what defines a local diffeomorphism but not a full diffeomorphism. Completing this with a formal statement:

Formal Definition

A map is invertible if it is 1:1 and onto. A map is locally invertible if for every point in the domain there is a neighborhood around that point which is 1:1 onto its image. That is, no nearby points take the same value, but distant points can. More precisely, there should not be points that take the same value arbitrarily close by; however, there can be a finite distance away.

Turning "invertible" into "diffeomorphism" simply requires that all maps be smooth with a smooth inverse.

Conclusion

The distinction between a diffeomorphism and a local diffeomorphism is critical in differential geometry. While every diffeomorphism is a local diffeomorphism, not every local diffeomorphism is a diffeomorphism. Understanding this difference is essential for various applications in topology, geometry, and related fields. By grasping these definitions and examples, you can better appreciate the intricacies of smooth mappings between manifolds.