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

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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 transition between different coordinate systems. 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 explore the implications of this extension on the smooth structure of the manifold.

Smooth Manifolds and Smooth Maps

A smooth manifold is a Hausdorff, second-countable topological space that is locally homeomorphic to Euclidean space. The smooth structure on a manifold is defined by a smooth atlas, which is a collection of charts that provide a smooth transition between different coordinate systems. 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. The transition functions between charts are smooth, meaning that they are infinitely differentiable.

A smooth map between two smooth manifolds is a map that is smooth on each chart. In other words, if f: M → N is a smooth map between two smooth manifolds M and N, then for each chart (U, φ) on M and each chart (V, ψ) on N, the composition ψ ∘ f ∘ φ^(-1) is smooth on the open subset φ(U) of Euclidean space.

Extension of a Smooth Map

Given a smooth map f: U → N between two smooth manifolds M and N, where U is an open subset of M, we want to extend f to a smooth map F: M → N on the whole of M. This means that we want to define a smooth map F on the entire manifold M, such that F = f on U.

Theorem 1: Existence of Extension

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

Proof

Let (U, φ) be a chart on M, and let (V, ψ) be a chart on N. Then, the transition function ψ ∘ f ∘ φ^(-1) is smooth on the open subset φ(U) of Euclidean space. Since U is an open subset of M, we can extend φ(U) to the entire manifold M by defining a new chart (M, φ') on M, where φ' is a homeomorphism from M to an open subset of Euclidean space. Then, the transition function ψ ∘ F ∘ φ'^(-1) is smooth on the entire manifold M.

Theorem 2: Uniqueness of Extension

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

Proof

Suppose that there are two extensions F and G of f to the whole of M. Then, F = f on U and G = f on U. Since F and G are smooth on the entire manifold M, we have F = G on M.

Corollary 1: Smooth Structure

Let f: U → N be a smooth map between two smooth manifolds M and N, where U is an open subset of M. Then, the extension F: M → N of f to the whole of M preserves the smooth structure of M.

Proof

Since F is smooth on the entire manifold M, we have that F is a diffeomorphism between M and N. Therefore, the smooth structure of M is preserved under the extension F.

Corollary 2: Smooth Maps

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

Proof

Since F is smooth on the entire manifold M, we have that F is a smooth map between M and N.

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 exists and is unique, and that it preserves the smooth structure of the manifold. We have also shown that the extension is a smooth map between the two manifolds. These results have important implications for the study of smooth manifolds and their properties.

References

  • Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry, revised 2nd edition. Academic Press.
  • Spivak, M. (1979). A comprehensive introduction to differential geometry, volume 1. Publish or Perish.
  • Warner, F. W. (1983). Foundations of differentiable manifolds and Lie groups. Springer-Verlag.

Further Reading

  • Lee, J. M. (2003). Introduction to smooth manifolds. Springer-Verlag.
  • Hirsch, M. W. (1976). Differential topology. Springer-Verlag.
  • Milnor, J. W. (1963). Morse theory. Princeton University Press.
    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 goal of extending a smooth map from an open subset of a manifold to the whole of the manifold?

A: The main goal of extending a smooth map from an open subset of a manifold to the whole of the manifold is to define a smooth map on the entire manifold, such that the map is smooth on each chart.

Q: What is the significance of the extension of a smooth map?

A: The extension of a smooth map is significant because it allows us to define a smooth map on the entire manifold, which is a fundamental concept in differential geometry. The extension of a smooth map also preserves the smooth structure of the manifold.

Q: What are the implications of the extension of a smooth map on the smooth structure of the manifold?

A: The extension of a smooth map preserves the smooth structure of the manifold, which means that the smooth structure of the manifold is preserved under the extension of the smooth map.

Q: What are the conditions for the existence of an extension of a smooth map?

A: The conditions for the existence of an extension of a smooth map are that the map must be smooth on each chart, and the transition functions between charts must be smooth.

Q: Is the extension of a smooth map unique?

A: Yes, the extension of a smooth map is unique. This means that if there are two extensions of a smooth map, then they must be equal.

Q: What are the properties of the extension of a smooth map?

A: The extension of a smooth map is a smooth map between the two manifolds, and it preserves the smooth structure of the manifold.

Q: Can the extension of a smooth map be used to define a smooth map on a larger manifold?

A: Yes, the extension of a smooth map can be used to define a smooth map on a larger manifold. This means that if we have a smooth map on a smaller manifold, we can extend it to a larger manifold using the extension of a smooth map.

Q: What are the applications of the extension of a smooth map in differential geometry?

A: The extension of a smooth map has many applications in differential geometry, including the study of smooth manifolds, Lie groups, and Riemannian geometry.

Q: Can the extension of a smooth map be used to define a smooth map on a non-compact manifold?

A: Yes, the extension of a smooth map can be used to define a smooth map on a non-compact manifold. This means that if we have a smooth map on a compact manifold, we can extend it to a non-compact manifold using the extension of a smooth map.

Q: What are the challenges of extending a smooth map from an open subset of a manifold to the whole of the manifold?

A: The challenges of extending a smooth map from an open subset of a manifold to the whole of the manifold include ensuring that the map is smooth on each chart, and that the transition functions between charts are smooth.

Q: Can the extension of a smooth map be used to define a smooth map on a manifold with boundary?

A: Yes, the extension of a smooth map can be used to define a smooth map on a manifold with boundary. This means that if we have a smooth map on a manifold with boundary, we can extend it to the whole manifold using the extension of a smooth map.

Q: What are the implications of the extension of a smooth map on the topology of the manifold?

A: The extension of a smooth map preserves the topology of the manifold, which means that the extension of a smooth map does not change the topological properties of the manifold.

Q: Can the extension of a smooth map be used to define a smooth map on a manifold with singularities?

A: Yes, the extension of a smooth map can be used to define a smooth map on a manifold with singularities. This means that if we have a smooth map on a manifold with singularities, we can extend it to the whole manifold using the extension of a smooth map.

Q: What are the applications of the extension of a smooth map in physics and engineering?

A: The extension of a smooth map has many applications in physics and engineering, including the study of classical mechanics, electromagnetism, and general relativity.

Q: Can the extension of a smooth map be used to define a smooth map on a manifold with non-trivial topology?

A: Yes, the extension of a smooth map can be used to define a smooth map on a manifold with non-trivial topology. This means that if we have a smooth map on a manifold with non-trivial topology, we can extend it to the whole manifold using the extension of a smooth map.