Smooth maps

Definition

Smooth maps are functions between differentiable manifolds that possess continuous derivatives of all orders. They play a crucial role in differential topology as they allow for the analysis of geometric and topological properties of manifolds through calculus. The concept of smoothness is essential for discussing transversality, as it ensures that the relevant structures behave nicely under perturbations.

5 Must Know Facts For Your Next Test

1. Smooth maps are defined as functions where all derivatives exist and are continuous, allowing them to be manipulated using techniques from calculus.
2. In the context of transversality, smooth maps help ensure that the intersection of submanifolds occurs with the expected dimensionality.
3. A smooth map between two manifolds can be locally expressed in terms of coordinates, making analysis more manageable.
4. The composition of two smooth maps is also a smooth map, which is essential for building more complex functions and understanding their properties.
5. Smooth maps can be used to define various important concepts in topology, such as homotopy and smooth homotopy equivalence.

Review Questions

• How do smooth maps facilitate the study of geometric properties in differential topology?
  • Smooth maps enable the application of calculus to study geometric properties on manifolds by ensuring that derivatives exist and behave continuously. This continuous behavior allows mathematicians to analyze how shapes deform and interact, particularly when investigating intersections and embeddings. The connection between smoothness and calculus is vital for understanding deeper concepts like transversality, where the intersection behavior of various structures becomes crucial.
• What role do smooth maps play in establishing transversality between two manifolds?
  • Smooth maps are central to establishing transversality because they ensure that the images of these maps intersect transversely. When two smooth maps are transversal, their images intersect in a manner that respects the expected dimensions, which is key to understanding how submanifolds relate within a manifold. This controlled intersection property is fundamental in many areas, including the study of critical points and stability within differential topology.
• Evaluate the implications of smoothness on the composition of functions in differential topology and its relevance to transversality.
  • The fact that the composition of smooth maps is itself a smooth map has significant implications in differential topology. It allows for the construction of more complex functions while retaining the desired properties necessary for analyzing geometric structures. In relation to transversality, this means one can create composed functions that maintain transversal intersection properties, thus enabling more intricate studies of how various manifolds interact and intersect under smooth transformations.

Related terms

Differentiable manifold: A topological space that locally resembles Euclidean space and allows for the definition of smooth functions, providing a setting for calculus on more complex shapes.

Transversality:

A property of two smooth maps that guarantees their images intersect in a controlled manner, crucial for understanding the intersections of submanifolds.

Tangent space: The vector space consisting of tangent vectors at a point on a manifold, providing a way to discuss derivatives and smooth maps at that point.