Sard's Theorem

Definition

Sard's Theorem states that the set of critical values of a smooth map between manifolds has measure zero. This means that, when mapping from one manifold to another, most points in the target manifold are regular values, which helps understand how the smooth map behaves. The implications of this theorem stretch into various areas, such as the properties of submersions and regular values, influencing the topology and geometry involved in differentiable maps.

5 Must Know Facts For Your Next Test

1. Sard's Theorem highlights that while critical points are important, they form a negligible set in terms of measure when considering the entire range of values in a smooth map.
2. The theorem applies to any smooth map between differentiable manifolds, regardless of their dimension or topology.
3. In practical applications, Sard's Theorem ensures that most values you might want to achieve with a smooth function can actually be obtained by some input from its domain.
4. The implications of Sard's Theorem are crucial for understanding transversality and how one manifold intersects another under a smooth mapping.
5. Sard's Theorem has profound effects in areas such as optimization and differential topology, providing insights into how functions behave near critical points.

Review Questions

• How does Sard's Theorem relate to the concepts of critical values and regular values in differential topology?
  • Sard's Theorem establishes that the set of critical values—those points in the target space where critical points from the domain map—has measure zero. This implies that most values in the target space are regular values. In other words, while critical points can be significant in analyzing a map's behavior, they are rare compared to regular values, reinforcing the idea that smooth maps are generally well-behaved and predictable almost everywhere.
• Discuss the implications of Sard's Theorem on submersions and how it influences our understanding of differentiable maps.
  • Sard's Theorem asserts that if a smooth map is a submersion at a point, then nearby points will also be regular values. This means that when working with submersions—where the differential is surjective at all points—we can ensure that most values we encounter will be non-critical. This relationship deepens our understanding of how differentiable maps can behave across different contexts, guiding us in applications like finding solutions to systems of equations.
• Evaluate how Sard's Theorem impacts advanced topics like transversality and degree theory in differential topology.
  • Sard's Theorem plays a crucial role in transversality, as it ensures that for most maps, intersections occur cleanly at regular values rather than at critical ones. This clean intersection property simplifies many problems in differential topology. Additionally, it influences degree theory by clarifying how many times a map wraps around a target manifold; since critical values are negligible, we can confidently analyze degrees without being bogged down by anomalies introduced by critical points.

Related terms

Critical Value: A point in the target space of a smooth map that is the image of a critical point in the domain, where the differential fails to be surjective.

Smooth Map: A function between differentiable manifolds that is infinitely differentiable, meaning it has continuous derivatives of all orders.

Regular Value: A point in the target space of a smooth map for which every pre-image is a regular point, meaning the differential of the map is surjective at those points.