Smooth Map

Definition

A smooth map is a function between two smooth manifolds that preserves the structure of the manifolds by ensuring that the map is infinitely differentiable. This means that when you take derivatives of the map, all of them exist, making it crucial in studying the relationships and properties of smooth manifolds. Smooth maps are foundational in understanding how different geometric structures interact and play a vital role in various advanced topics like submersions and embeddings.

5 Must Know Facts For Your Next Test

1. Smooth maps are defined between two smooth manifolds and must be infinitely differentiable on their domains.
2. The composition of smooth maps is also a smooth map, which helps build complex structures from simpler ones.
3. Smooth maps can also be used to define concepts such as embeddings and immersions, which relate to how manifolds sit inside one another.
4. In the context of Riemannian geometry, smooth maps preserve the Riemannian structure, allowing for comparisons of distances and angles.
5. Smooth maps play a key role in defining geometric structures like vector fields and differential forms on manifolds.

Review Questions

• How does the property of being infinitely differentiable impact the study of relationships between smooth manifolds?
  • The requirement that smooth maps are infinitely differentiable ensures that not only can we compute rates of change but also analyze how different manifolds relate to each other. This property allows us to apply calculus tools effectively, enabling us to derive critical insights into the geometric and topological features shared between the manifolds. The preservation of structure through these smooth maps is essential for constructing complex mathematical frameworks in differential geometry.
• What is the significance of smooth maps in the context of Riemannian submersions and how do they contribute to understanding these structures?
  • In Riemannian submersions, smooth maps serve as foundational elements that define how one manifold smoothly projects onto another while preserving certain geometric properties. They facilitate the analysis of distance and angle relationships in lower-dimensional spaces while retaining the essential features of the upper-dimensional manifold. The behavior of smooth maps in this context allows for applications such as understanding geodesics and curvature in various geometrical settings.
• Evaluate how smooth maps influence the classification of embedded and immersed submanifolds and their importance in differential geometry.
  • Smooth maps are critical in classifying embedded and immersed submanifolds as they determine how these submanifolds interact with their ambient spaces. When an embedding is defined via a smooth map, it ensures that local properties are preserved, allowing for meaningful comparisons between different geometrical entities. Furthermore, this classification aids in simplifying complex problems by breaking them down into manageable parts while enriching our understanding of curvature, topology, and other essential characteristics inherent in differential geometry.

Related terms

Differentiable Function: A function that has a derivative at every point in its domain, meaning it is continuous and its rate of change can be determined.

Manifold:

A topological space that locally resembles Euclidean space and allows for concepts from calculus to be applied.

Submersion:

A type of smooth map where the differential of the map is surjective, indicating that it locally projects points down to a lower-dimensional space.