Partition of unity

Definition

A partition of unity is a collection of continuous functions on a manifold that are used to define a global object from local data. Each function in this collection is non-negative and locally supported, meaning it is zero outside a certain compact subset. Partitions of unity allow mathematicians to work with objects defined in local neighborhoods and piece them together coherently over the entire manifold.

5 Must Know Facts For Your Next Test

1. Partitions of unity are essential in differential geometry for extending local constructions to global ones across manifolds.
2. Each function in a partition of unity sums to one when evaluated at any point in the manifold, ensuring that they can combine local information effectively.
3. The functions in a partition of unity can be chosen to be smooth (infinitely differentiable), making them very useful for calculus on manifolds.
4. The existence of partitions of unity relies on the ability to cover the manifold with locally finite open covers.
5. Using bump functions, which are smooth and compactly supported, allows for a constructive way to create partitions of unity.

Review Questions

• How does a partition of unity enable the combination of local data into a global context?
  • A partition of unity allows mathematicians to take local information from neighborhoods within a manifold and combine it into a global framework. Each function in the partition is associated with a local patch, contributing to the overall structure while maintaining control over local properties. By ensuring that these functions are non-negative and sum up to one, they can effectively manage how much influence each local piece has in the global picture.
• Discuss how bump functions are used in constructing partitions of unity and their properties.
  • Bump functions play a crucial role in creating partitions of unity because they are smooth and compactly supported. This means they are zero outside a certain region, which makes them perfect for local constructions. When constructing a partition of unity, bump functions can be adjusted so that they contribute to the partition's requirement of being non-negative and summing up to one. The smoothness ensures that any resulting global object retains desirable properties, such as differentiability.
• Evaluate the importance of locally finite covers in the existence of partitions of unity on manifolds and how this impacts differential topology.
  • Locally finite covers are vital for the existence of partitions of unity because they ensure that at any point on the manifold, only finitely many sets from the cover intersect any neighborhood. This property simplifies the construction process, allowing for each piece of local data to be blended smoothly into the whole. In differential topology, this leads to powerful results such as extending functions and forms across manifolds, making it easier to analyze complex structures by leveraging local insights.

Related terms

Locally finite: A collection of sets is locally finite if every point in the space has a neighborhood that intersects only finitely many sets from the collection.

Bump function: A smooth function that is compactly supported, meaning it is non-zero only on a limited set, commonly used to construct partitions of unity.

Manifold: A topological space that locally resembles Euclidean space and allows for the definition of concepts like continuity, differentiability, and integration.