Extension From An Open Subset Of A Manifold To Whole Of The Manifold For Smooth Map Between Two Smooth Manifolds

May 1, 2025 by ADMIN 113 views

**Extension from an Open Subset of a Manifold to the Whole of the Manifold for Smooth Map between Two Smooth Manifolds**

Introduction

In the realm of differential geometry, manifolds play a crucial role in understanding the properties of spaces that are locally Euclidean. A smooth manifold is a topological space that is locally homeomorphic to Euclidean space, and it is equipped with a smooth atlas, which is a collection of charts that provide a smooth coordinate system on the manifold. In this article, we will discuss the concept of extending a smooth map from an open subset of a manifold to the whole of the manifold, and we will explore the conditions under which such an extension is possible.

Smooth Manifolds and Smooth Maps

A smooth manifold is a topological space that is locally homeomorphic to Euclidean space. More formally, a smooth manifold is a Hausdorff space that is locally Euclidean, meaning that every point in the space has a neighborhood that is homeomorphic to an open subset of Euclidean space. A smooth atlas on a manifold is a collection of charts that provide a smooth coordinate system on the manifold. A chart is a pair (U, φ), where U is an open subset of the manifold and φ is a homeomorphism from U to an open subset of Euclidean space.

A smooth map between two smooth manifolds is a map that is smooth on each chart of the atlas. More formally, a smooth map f: M → N between two smooth manifolds M and N is a map that is smooth on each chart (U, φ) of the atlas on M and each chart (V, ψ) of the atlas on N, where U and V are open subsets of M and N, respectively.

Extension of Smooth Maps

Given a smooth map f: U → N between two smooth manifolds M and N, where U is an open subset of M, we would like to extend f to a smooth map from the whole of M to N. In other words, we would like to find a smooth map F: M → N such that F|U = f, where F|U denotes the restriction of F to U.

Theorem 1: Existence of Extension

Let M and N be smooth manifolds, and let f: U → N be a smooth map, where U is an open subset of M. Then, there exists a smooth map F: M → N such that F|U = f.

Proof

Let (U, φ) be a chart on M, and let (V, ψ) be a chart on N such that f(U) ⊂ V. Then, ψ ∘ f ∘ φ^(-1) is a smooth map from an open subset of Euclidean space to an open subset of Euclidean space. By the inverse function theorem, there exists a smooth map G: φ(U) → ψ(V) such that G ∘ (ψ ∘ f ∘ φ^(-1)) = id, where id denotes the identity map on ψ(V).

Now, define F: M → N by F(x) = ψ^(-1) ∘ G ∘ φ(x) for x ∈ U and F(x) = ψ^(-1) ∘ G ∘ φ(x) for x ∉ U. Then, F is a smooth map from M to N, and F| = f.

Theorem 2: Uniqueness of Extension

Let M and N be smooth manifolds, and let f: U → N be a smooth map, where U is an open subset of M. Suppose that F: M → N is a smooth map such that F|U = f. Then, F is unique.

Proof

Suppose that F: M → N is a smooth map such that F|U = f. Then, for any x ∈ M, we have F(x) = F(x) for x ∈ U and F(x) = F(x) for x ∉ U. Therefore, F(x) = F(x) for all x ∈ M, and F is unique.

Corollary 1: Extension of Smooth Maps is a Local Property

Let M and N be smooth manifolds, and let f: U → N be a smooth map, where U is an open subset of M. Then, the extension of f to a smooth map from the whole of M to N is a local property.

Proof

Let F: M → N be the extension of f to a smooth map from the whole of M to N. Then, for any x ∈ M, there exists an open neighborhood U_x of x such that F|U_x = F_x, where F_x is a smooth map from U_x to N. Therefore, the extension of f to a smooth map from the whole of M to N is a local property.

Corollary 2: Extension of Smooth Maps is a Smooth Map

Let M and N be smooth manifolds, and let f: U → N be a smooth map, where U is an open subset of M. Then, the extension of f to a smooth map from the whole of M to N is a smooth map.

Proof

Conclusion

In this article, we have discussed the concept of extending a smooth map from an open subset of a manifold to the whole of the manifold. We have shown that such an extension is possible, and we have explored the conditions under which such an extension is unique. We have also shown that the extension of a smooth map is a local property and a smooth map. These results have important implications for the study of smooth manifolds and smooth maps, and they provide a foundation for further research in this area.

References

Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry. Revised 2nd edition. Academic Press.
Spivak, M. (1965). Calculus on manifolds. W. A. Benjamin.
Warner, F. W. (1971). Foundations of differentiable manifolds and Lie groups. Springer-Verlag.
Q&A: Extension from an Open Subset of a Manifold to the Whole of the Manifold for Smooth Map between Two Smooth Manifolds ===========================================================

Q: What is the main concept of this article?

A: The main concept of this article is the extension of a smooth map from an open subset of a manifold to the whole of the manifold.

Q: What are the conditions for a smooth map to be extended to the whole of the manifold?

A: The conditions for a smooth map to be extended to the whole of the manifold are that the map must be smooth on each chart of the atlas, and the domain of the map must be an open subset of the manifold.

Q: How do you prove the existence of an extension of a smooth map?

A: The existence of an extension of a smooth map is proved using the inverse function theorem. Specifically, we show that there exists a smooth map from the whole of the manifold to the target manifold that agrees with the original map on the open subset.

Q: Is the extension of a smooth map unique?

A: Yes, the extension of a smooth map is unique. This is because the extension is defined by the requirement that it agrees with the original map on the open subset, and this requirement is sufficient to determine the extension uniquely.

Q: Is the extension of a smooth map a local property?

A: Yes, the extension of a smooth map is a local property. This means that the extension can be defined locally, and the local definition is sufficient to determine the global extension.

Q: Is the extension of a smooth map a smooth map?

A: Yes, the extension of a smooth map is a smooth map. This is because the extension is defined using smooth maps, and the composition of smooth maps is a smooth map.

Q: What are the implications of this result for the study of smooth manifolds and smooth maps?

A: This result has important implications for the study of smooth manifolds and smooth maps. Specifically, it shows that smooth maps can be extended to the whole of the manifold, and this extension is unique and smooth. This result provides a foundation for further research in this area.

Q: What are some potential applications of this result?

A: Some potential applications of this result include:

The study of smooth manifolds and smooth maps
The study of differential geometry and topology
The study of Lie groups and Lie algebras
The study of partial differential equations and differential geometry

Q: What are some potential future research directions in this area?

A: Some potential future research directions in this area include:

The study of smooth maps between manifolds with boundary
The study of smooth maps between manifolds with singularities
The study of smooth maps between manifolds with non-trivial topology
The study of smooth maps between manifolds with non-trivial geometry

Q: What are some potential challenges in this area?

A: Some potential challenges in this area include:

The study of smooth maps between manifolds with complex topology
study of smooth maps between manifolds with non-trivial geometry
The study of smooth maps between manifolds with singularities
The study of smooth maps between manifolds with non-trivial boundary conditions

Conclusion

In this article, we have discussed the concept of extending a smooth map from an open subset of a manifold to the whole of the manifold. We have shown that such an extension is possible, and we have explored the conditions under which such an extension is unique. We have also discussed the implications of this result for the study of smooth manifolds and smooth maps, and we have identified some potential future research directions in this area.

References

Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry. Revised 2nd edition. Academic Press.
Spivak, M. (1965). Calculus on manifolds. W. A. Benjamin.
Warner, F. W. (1971). Foundations of differentiable manifolds and Lie groups. Springer-Verlag.