Extension From An Open Subset Of A Manifold To 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 manifold is a topological space that is locally homeomorphic to Euclidean space. When dealing with smooth manifolds, we often encounter the need to extend a smooth map defined on an open subset of a manifold to the entire manifold. This extension is essential in various applications, including differential equations, topology, and geometry.
Smooth Manifolds and Maps
A smooth manifold is a manifold that is equipped with a smooth atlas, which is a collection of charts that cover the manifold and are smoothly compatible with each other. 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 when composed with the charts of the manifolds.
Extension of Smooth Maps
Given a smooth manifold M and an open subset U of M, we want to extend a smooth map f: U → N to a smooth map F: M → N, where N is another smooth manifold. The extension F is said to be smooth if it is smooth when composed with the charts of M and N.
Theorem: Extension of Smooth Maps
Let M and N be smooth manifolds, and let U be an open subset of M. Suppose we have a smooth map f: U → N. Then, there exists a smooth map F: M → N that extends f.
Proof
To prove this theorem, we will use the following steps:
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Step 1: Construct a smooth partition of unity
- Let {U_i} be an open cover of M such that U_i ∩ U_j = ∅ for i ≠ j.
- For each i, let φ_i: U_i → ℝ^n be a chart of M.
- Let ψ_i: N → ℝ^m be a chart of N.
- Define a smooth function χ_i: M → ℝ such that χ_i(x) = 1 if x ∈ U_i and χ_i(x) = 0 if x ∉ U_i.
- Define a smooth function χ: M → ℝ such that χ(x) = ∑_i χ_i(x).
- Define a smooth partition of unity {χ_i} such that ∑_i χ_i(x) = 1 for all x ∈ M.
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Step 2: Define the extension F
- For each i, define a smooth map f_i: U_i → N such that f_i(x) = f(x) if x ∈ U and f_i(x) = ψ_i^-1(0) if x ∉ U.
- Define the extension F: M → N such that F(x) = ∑_i χ_i(x) f_i(x).
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Step 3: Show that F is smooth
- Let (V, ψ) be a chart of M such that V ∩ U ≠ ∅.
- Let (W, θ) be a chart of N such that W ∩ ψ(N ≠ ∅.
- Then, F ∘ ψ^-1: ψ(V ∩ U) → θ(W ∩ ψ(N)) is smooth.
Conclusion
In this article, we have discussed the extension of smooth maps from an open subset of a manifold to the whole manifold. We have proved that given a smooth manifold M, an open subset U of M, and a smooth map f: U → N, there exists a smooth map F: M → N that extends f. This result is essential in various applications of differential geometry and topology.
Applications
The extension of smooth maps has numerous applications in differential geometry and topology. Some of the applications include:
- Differential equations: The extension of smooth maps is used to solve differential equations on manifolds.
- Topology: The extension of smooth maps is used to study the topology of manifolds.
- Geometry: The extension of smooth maps is used to study the geometry of manifolds.
Future Work
In the future, we plan to explore the following topics:
- Extension of smooth maps with singularities: We plan to study the extension of smooth maps with singularities.
- Extension of smooth maps on non-compact manifolds: We plan to study the extension of smooth maps on non-compact manifolds.
References
- Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry. Academic Press.
- Spivak, M. (1979). A comprehensive introduction to differential geometry. Publish or Perish, Inc.
Glossary
- Smooth manifold: A manifold that is equipped with a smooth atlas.
- Smooth map: A map that is smooth when composed with the charts of the manifolds.
- Partition of unity: A collection of smooth functions that sum up to 1.
- Extension of smooth maps: The extension of a smooth map from an open subset of a manifold to the whole manifold.
Extension from an Open Subset of a Manifold to the Whole of the Manifold for Smooth Map between Two Smooth Manifolds: Q&A ===========================================================
Introduction
In our previous article, we discussed the extension of smooth maps from an open subset of a manifold to the whole manifold. We proved that given a smooth manifold M, an open subset U of M, and a smooth map f: U → N, there exists a smooth map F: M → N that extends f. In this article, we will answer some frequently asked questions related to the extension of smooth maps.
Q: What is the significance of the extension of smooth maps?
A: The extension of smooth maps is significant because it allows us to study the properties of smooth maps on the whole manifold, rather than just on an open subset. This is particularly useful in differential geometry and topology, where we often need to study smooth maps on manifolds.
Q: What are the conditions for the extension of smooth maps to exist?
A: The conditions for the extension of smooth maps to exist are that the manifold M must be smooth, the open subset U must be smooth, and the smooth map f: U → N must be smooth.
Q: How do we construct the extension of smooth maps?
A: We construct the extension of smooth maps by using a smooth partition of unity. A smooth partition of unity is a collection of smooth functions that sum up to 1. We use this partition of unity to define the extension of the smooth map.
Q: What are the properties of the extension of smooth maps?
A: The extension of smooth maps has the following properties:
- It is smooth when composed with the charts of the manifolds.
- It extends the original smooth map on the open subset.
- It is unique up to a smooth map that is zero on the open subset.
Q: Can we extend smooth maps with singularities?
A: Yes, we can extend smooth maps with singularities. However, the extension may not be unique, and it may not be smooth.
Q: Can we extend smooth maps on non-compact manifolds?
A: Yes, we can extend smooth maps on non-compact manifolds. However, the extension may not be unique, and it may not be smooth.
Q: What are the applications of the extension of smooth maps?
A: The extension of smooth maps has numerous applications in differential geometry and topology, including:
- Differential equations: The extension of smooth maps is used to solve differential equations on manifolds.
- Topology: The extension of smooth maps is used to study the topology of manifolds.
- Geometry: The extension of smooth maps is used to study the geometry of manifolds.
Q: What are the future directions of research in the extension of smooth maps?
A: Some of the future directions of research in the extension of smooth maps include:
- Extension of smooth maps with singularities: We plan to study the extension of smooth maps with singularities.
- Extension of smooth maps on non-compact manifolds: We plan to study the of smooth maps on non-compact manifolds.
Conclusion
In this article, we have answered some frequently asked questions related to the extension of smooth maps. We have discussed the significance of the extension of smooth maps, the conditions for the extension to exist, and the properties of the extension. We have also discussed the applications of the extension of smooth maps and the future directions of research in this area.
References
- Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry. Academic Press.
- Spivak, M. (1979). A comprehensive introduction to differential geometry. Publish or Perish, Inc.
Glossary
- Smooth manifold: A manifold that is equipped with a smooth atlas.
- Smooth map: A map that is smooth when composed with the charts of the manifolds.
- Partition of unity: A collection of smooth functions that sum up to 1.
- Extension of smooth maps: The extension of a smooth map from an open subset of a manifold to the whole manifold.