1 Definition and basic concepts
A regular value is a point in the target of a differentiable map for which the map has maximal local rank along every point of the corresponding fiber. In differential topology, regular values are important because their level sets typically have the most tractable geometric form: they are smooth sets whose dimension can be read directly from the rank of the derivative. This makes them a basic tool for studying equations, constraints, and intersections on manifolds.
1.1 Differentiable maps
The notion of regular value applies to differentiable maps between smooth manifolds or, more simply, between open subsets of Euclidean spaces. Such a map assigns each point in the domain to a point in the codomain and has a derivative at each point. The derivative is a linear approximation to the map near that point, and its rank reflects how much local information the map preserves.
In the manifold setting, differentiability is defined using coordinate charts, but the basic idea remains the same. The map must be smooth enough that its first derivative exists and varies continuously when needed for standard results.
1.2 Critical points and critical values
A point in the domain is called a critical point if the derivative at that point fails to have full rank. The image of a critical point is a critical value. These are the target values where the map may have singular or degenerate level sets.
Critical points mark places where the map loses local regularity. In geometric applications, they often correspond to folds, cusps, or other singular features. The set of critical values may be small in a measure-theoretic sense, yet it often carries important geometric information.
1.3 Regular points and regular values
A regular point is a point in the domain where the derivative has maximal possible rank. A regular value is a point in the codomain such that every point in its preimage is regular. This distinction is central: a value is regular not because it is special by itself, but because all points mapping to it behave uniformly well.
1.3.1 Surjective derivative condition
For a map between manifolds, a value is regular when the derivative at each preimage point is surjective onto the tangent space of the codomain. In Euclidean spaces, this means the Jacobian matrix has full rank equal to the dimension of the target. Surjectivity ensures that the map locally reaches all target directions, which prevents degeneracy in the corresponding level set.
1.3.2 Preimage of a value
The preimage of a value is the set of all points in the domain mapped to that value. When the value is regular, this preimage often forms a smooth submanifold of the domain. If the preimage is empty, the value is sometimes treated as regular by convention, since the defining condition is vacuously satisfied.
1.4 Examples of regular and critical values
For the function \(f(x)=x^2\) from the real line to itself, every nonnegative number is in the image, but \(0\) is a critical value because the derivative vanishes at \(x=0\). Any positive number is a regular value, since its preimages are points where the derivative is nonzero.
For the distance function from the plane to the origin, nonzero radii are regular values, and the corresponding level sets are circles. The origin is a critical value because the derivative degenerates at the center.
2 Regular value theorem
The regular value theorem is one of the most useful structural results in differential topology. It explains why level sets of regular values are smooth objects and gives their dimension directly from the dimensions of the domain and codomain.
2.1 Statement of the theorem
If \(f : M \to N\) is a smooth map between smooth manifolds and \(y \in N\) is a regular value, then the preimage \(f^{-1}(y)\) is a smooth submanifold of \(M\). Its dimension equals \(\dim M - \dim N\) when \(f\) is a submersion at every point of the fiber.
In Euclidean language, if a smooth map from \(\mathbb{R}^m\) to \(\mathbb{R}^n\) has \(y\) as a regular value, then the solution set \(f^{-1}(y)\) is locally a smooth manifold of dimension \(m-n\).
2.2 Dimension of the level set
The theorem gives a precise dimension formula for regular level sets. The rank of the derivative tells how many independent constraints the equation \(f(x)=y\) imposes. If the derivative is surjective, then the constraints are independent, and the resulting solution set has the expected dimension.
This principle is especially valuable in geometry and analysis, where one wants to know whether a system of equations defines a curve, surface, or higher-dimensional manifold rather than a singular set.
2.3 Relationship to submanifolds
Regular level sets are not just abstract sets; they inherit a smooth manifold structure from the ambient space. This makes them examples of embedded submanifolds, often described locally by equations with full-rank derivatives.
2.3.1 Embedded submanifolds
An embedded submanifold is a subset that sits inside a larger manifold in a smooth and well-behaved way. Regular level sets provide a standard source of such submanifolds. They are defined by equations, but near each point they look like coordinate subspaces after a suitable change of variables.
2.3.2 Local coordinate description
Near a regular point, coordinates can be chosen so that the map takes a simple form, often resembling projection onto the first few coordinates. In this local picture, the level set becomes a slice defined by setting some coordinates equal to constants. This coordinate normal form makes the manifold structure transparent.
3 Geometric and topological significance
Regular values reveal the geometric shape of solution sets and explain when those sets are stable under small perturbations. They are also useful in topology, since the structure of a level set can influence the global behavior of a manifold or mapping.
3.1 Level sets as manifolds
A level set is the collection of points where a function takes the same value. When the value is regular, the level set is smooth and has no singular points caused by rank deficiency. This makes regular level sets ideal for geometric study, since tools from manifold theory apply directly.
3.2 Tangent spaces of regular level sets
At a point in a regular level set, the tangent space consists of vectors annihilated by the derivative of the map. In other words, it is the kernel of the differential at that point. This gives a clean linear description of the directions along which one can move while staying inside the level set.
3.3 Implications for smooth structure
Because regular level sets are manifolds, they carry tangent spaces, charts, and other smooth structures. This allows one to integrate, orient, and study curvature on them when additional conditions are met. The regular value condition therefore serves as a gateway from equations to geometry.
4 Applications in differential topology
Regular values appear throughout differential topology, where they help convert analytic conditions into geometric ones. They are especially important in the study of maps between manifolds and in arguments that rely on genericity.
4.1 Preimage theorem
The preimage theorem is a broad formulation of the regular value theorem. It states that the preimage of a submanifold under a map transverse to that submanifold is itself a submanifold. Regular values are the special case where the target submanifold is a single point.
This theorem is widely used to prove that intersections and constraint sets have the expected dimension, provided the relevant transversality hypotheses hold.
4.2 Transversality
Transversality is a condition describing how one geometric object meets another. When a map is transverse to a submanifold, its preimage inherits smooth manifold structure. Regular values correspond to transversality with points, so they form the simplest instance of this idea.
The concept is powerful because transversality is often stable under perturbation. As a result, many geometric constructions can be arranged to produce regular values or transverse intersections.
4.3 Sard's theorem
Sard's theorem states that the set of critical values of a smooth map is small in a precise sense. This result underlies the practical usefulness of regular values, since it implies that regular values are abundant.
4.3.1 Density of regular values
In many standard settings, regular values are dense in the target. This means that even if a specific value is singular, nearby values often behave better. Density is crucial when one wants to perturb a problem slightly to obtain a smooth solution set.
4.3.2 Measure-theoretic consequences
Sard's theorem implies that the set of critical values has measure zero under appropriate smoothness assumptions. Consequently, almost every value is regular in the measure-theoretic sense. This makes regular values a generic phenomenon rather than an exceptional one.
5 Examples and computations
Concrete examples help show how regular values are identified in practice. The key step is usually to compute the derivative and check whether it has full rank on the relevant preimage.
5.1 Functions between Euclidean spaces
For a smooth function \(f:\mathbb{R}^m \to \mathbb{R}^n\), one computes the Jacobian matrix. A value \(y\) is regular if, at every point \(x\) with \(f(x)=y\), the Jacobian has rank \(n\). If the rank drops at any such point, then \(y\) is critical.
Typical examples include polynomial maps, trigonometric maps, and coordinate projections. In each case, the geometry of the fiber reflects the rank of the derivative.
5.2 Maps on manifolds
On manifolds, one uses local coordinates to reduce the problem to Euclidean computation. A map may be regular at a value even when its formula looks complicated globally, provided the local derivative is surjective along the entire fiber. This is common in geometric constructions such as height functions, distance functions, and bundle projections.
5.3 Regular values in one-dimensional cases
For maps from an interval or the real line to the real line, regular values are those whose preimages contain only points where the derivative is nonzero. In this setting, regular values are easy to visualize: the graph crosses the horizontal line at nonflat points.
If a value is attained at a local maximum or minimum, it is often critical. By contrast, values attained on monotone parts of the graph are usually regular.
5.4 Regular values for projection maps
Projection maps often provide the simplest examples of regular values. A coordinate projection from a product manifold onto one factor is typically a submersion, so every value in the target is regular. The fibers are then the complementary factor, viewed as parallel copies inside the product.
These examples illustrate the general pattern: when a map loses no dimensions in the target direction, its level sets are especially well behaved.
6 Related concepts
Several closely related notions help place regular values in a broader framework. These ideas describe the behavior of individual points, whole fibers, and maps themselves.
6.1 Regular point
A regular point is a domain point where the derivative has maximal rank. Regular values are defined using regular points in the entire preimage. Thus the concept of a regular value is a fiberwise condition built from pointwise regularity.
6.2 Critical submanifold
In some settings, the set of critical points may itself form a submanifold. Such a critical submanifold often signals a structured failure of regularity rather than an isolated singularity. It can appear in Morse-Bott theory and related geometric frameworks.
6.3 Submersion
A submersion is a smooth map whose derivative is surjective at every point. Every value of a submersion is regular. Submersions therefore represent the strongest and most uniform case of regularity for maps between manifolds.
6.4 Immersion
An immersion is a smooth map whose derivative is injective at every point. Although the definition is dual in spirit to that of a submersion, it addresses embedding behavior rather than level sets. Immersions and submersions are often studied together as basic rank conditions in differential geometry.
7 Generalizations
The notion of regular value extends beyond finite-dimensional smooth manifolds. In more advanced settings, the same intuition persists, though technical hypotheses become more delicate.
7.1 Infinite-dimensional settings
In infinite-dimensional geometry, one studies maps between spaces such as function spaces or spaces of sections. Regular values can still be defined using derivatives between appropriate topological vector spaces, but the standard finite-dimensional theorems may require stronger analytic assumptions. Issues of completeness and local model structure become more prominent.
7.2 Banach and Hilbert manifolds
For Banach and Hilbert manifolds, versions of the regular value theorem continue to hold under suitable hypotheses. These results are central in nonlinear analysis and global analysis, where solution sets to partial differential equations are treated geometrically. The surjectivity of the derivative often must be supplemented by a splitting or Fredholm condition.
7.3 Smooth maps with boundary
When manifolds have boundary, regular value theory can be adapted to account for boundary points. The preimage of a regular value may meet the boundary in a controlled way, producing a manifold with boundary or corners. Such extensions are useful in cobordism theory and in geometric problems involving constrained domains.