5.3 Submersions and regular values
3 min read•august 9, 2024
Submersions and regular values are key concepts in differential topology. They help us understand how smooth maps behave between manifolds, focusing on surjective differentials and their implications for the structure of the map.
These ideas are crucial for analyzing quotient manifolds, foliations, and fiber bundles. They provide tools to study how manifolds can be projected onto lower-dimensional spaces, preserving certain properties in the process.
Submersions and Regular Values
Defining Submersions and Their Properties
Top images from around the web for Defining Submersions and Their Properties
calculus - The proof of inverse function theorem - two questions - Mathematics Stack Exchange View original
Is this image relevant?
Category:Differential topology - Wikimedia Commons View original
Is this image relevant?
Derivatives of Inverse Functions · Calculus View original
Is this image relevant?
calculus - The proof of inverse function theorem - two questions - Mathematics Stack Exchange View original
Is this image relevant?
Category:Differential topology - Wikimedia Commons View original
Is this image relevant?
1 of 3
Top images from around the web for Defining Submersions and Their Properties
calculus - The proof of inverse function theorem - two questions - Mathematics Stack Exchange View original
Is this image relevant?
Category:Differential topology - Wikimedia Commons View original
Is this image relevant?
Derivatives of Inverse Functions · Calculus View original
Is this image relevant?
calculus - The proof of inverse function theorem - two questions - Mathematics Stack Exchange View original
Is this image relevant?
Category:Differential topology - Wikimedia Commons View original
Is this image relevant?
1 of 3
- describes a smooth map between manifolds where the differential is surjective at every point
- means the linear map between tangent spaces is onto, ensuring the map covers the entire codomain
- refers to a point in the codomain where the differential of the map is surjective
- denotes a point in the codomain that is not a regular value, indicating potential singularities or complexities in the map
- Submersions exhibit local behavior similar to projections, allowing for simplified analysis in certain regions
- applies to submersions, guaranteeing local invertibility under specific conditions
Applications and Implications of Submersions
- Submersions play a crucial role in studying manifold structures and their relationships
- Used to define and analyze quotient manifolds, where the submersion map identifies points to be collapsed
- Facilitate the study of foliations, partitioning manifolds into submanifolds of lower
- Enable the construction of fiber bundles, geometric structures consisting of a total space, base space, and fibers
- Provide a framework for understanding how manifolds can be "flattened" or projected onto lower-dimensional spaces
- Submersions preserve certain topological and geometric properties between the domain and codomain
Level Sets and Preimages
Understanding Level Sets in Manifold Theory
- represents the set of all points in the domain that map to a specific value in the codomain
- Forms a submanifold of the domain under certain conditions (regular value theorem)
- states that the preimage of a regular value of a smooth map is a submanifold
- Dimension of the level set equals the difference between the dimensions of the domain and codomain
- Level sets provide a way to visualize and analyze the behavior of smooth maps between manifolds
- Used in various applications, including optimization problems and the study of dynamical systems
Exploring Fiber Bundles and Their Structure
- consists of a total space, base space, and fibers, connected by a
- Locally resembles a product space, but may have a more complex global structure
- Fibers represent the preimages of points in the base space under the projection map
- Examples include tangent bundles (fibers are tangent spaces) and vector bundles (fibers are vector spaces)
- describe how fibers are "glued" together over different regions of the base space
- Fiber bundles generalize the concept of cartesian products, allowing for twisted or non-trivial global structures
Sard's Theorem
Implications and Applications of Sard's Theorem
- states that the set of critical values of a smooth map has measure zero in the codomain
- Provides a powerful tool for analyzing the behavior of smooth maps between manifolds
- Implies that "most" points in the codomain are regular values, simplifying many theoretical arguments
- Used in proving the existence of to continuous functions
- Applies to infinite-dimensional manifolds under certain conditions, extending its utility to functional analysis
- Plays a crucial role in the study of , relating critical points of functions to the topology of manifolds
Understanding Regular and Critical Values
- Regular value occurs when the differential of the map is surjective at all points in its preimage
- Critical value happens when the differential fails to be surjective at least one point in its preimage
- Set of regular values forms an open dense subset of the codomain, ensuring their abundance
- Critical values may correspond to singularities, extrema, or other interesting features of the map
- Analyzing the behavior near critical values often reveals important information about the map and underlying manifolds
- Morse theory uses critical points and values to study the topology of manifolds through smooth functions
Key Terms to Review (23)
Critical Value: A critical value refers to a point in the domain of a function where its derivative is either zero or undefined, indicating potential maxima, minima, or points of inflection. These values are essential in understanding the behavior of functions, especially when analyzing smooth mappings between manifolds, as they help identify where the function fails to be a submersion. Additionally, critical values play a significant role in Sard's Theorem, which deals with the measure of sets of critical values and their implications on the image of functions.
Differentiable Map: A differentiable map is a function between two differentiable manifolds that is smooth, meaning it has continuous derivatives of all orders. This concept is essential in understanding how different manifolds relate to each other, particularly in terms of structure and behavior, as it allows for the analysis of curves, surfaces, and higher-dimensional spaces. Differentiable maps are foundational in the study of submersions, immersions, and the construction of product and quotient manifolds, where the nature of smoothness and the behavior of derivatives play a crucial role.
Dimension: Dimension refers to the minimum number of coordinates needed to specify a point within a mathematical space. It serves as a fundamental concept in topology and geometry, allowing us to classify spaces based on their complexity and structure. The concept of dimension connects various important features, such as the behavior of submanifolds, the intricacies of embeddings, and the properties of different types of manifolds like spheres and tori.
Fiber Bundle: A fiber bundle is a mathematical structure that consists of a base space, a total space, and a typical fiber such that locally, the total space looks like a product of the base space and the fiber. This concept is important for understanding how different spaces can be related to each other, especially in the context of submersions and regular values, as it allows us to analyze the behavior of continuous maps and their fibers, shedding light on the topology of spaces involved.
Foliation: Foliation is a geometric structure on a manifold that decomposes it into a collection of disjoint, smoothly varying submanifolds called leaves. Each leaf represents a local model of the manifold, allowing for the study of how these layers interact and align with each other. The concept is crucial when analyzing the behavior of functions, particularly in relation to submersions and regular values, as it helps in understanding how spaces can be broken down into simpler pieces.
Henri Poincaré: Henri Poincaré was a French mathematician and physicist, often regarded as one of the founders of topology and a pioneer in the study of dynamical systems. His work laid foundational concepts that connect with various branches of mathematics, especially in understanding the behavior of continuous functions and spaces.
Inverse Function Theorem: The Inverse Function Theorem states that if a function is continuously differentiable and its derivative is non-zero at a point, then it has a continuous inverse function near that point. This theorem plays a crucial role in understanding the behavior of smooth maps and their properties, as it provides conditions under which we can locally reverse mappings between spaces.
John Milnor: John Milnor is a prominent American mathematician known for his significant contributions to differential topology, particularly in the areas of manifold theory, Morse theory, and the topology of high-dimensional spaces. His work has fundamentally shaped the field and has broad implications for various topics within topology, including submersions, critical values, and cohomology groups.
Level Set: A level set is a set of points in a space where a given function takes on a constant value. This concept is vital in understanding how functions behave and can be visualized geometrically as surfaces or curves in higher dimensions. Level sets help in analyzing the properties of functions, particularly when dealing with implicit relations and the behavior of mappings, especially in contexts like local parameterizations and submersions.
Morse theory: Morse theory is a branch of mathematics that studies the topology of manifolds using smooth functions, particularly focusing on the critical points of these functions and their implications for the manifold's structure. By analyzing how these critical points behave under variations of the function, Morse theory connects the geometry of the manifold with its topology, providing deep insights into the shape and features of the space.
Preimage Theorem: The Preimage Theorem states that for a smooth map between manifolds, the preimage of a regular value is a submanifold of the domain manifold. This theorem is crucial in understanding how certain structures in differential topology are preserved under smooth maps, leading to insights about the topology of manifolds and their mappings.
Projection Map: A projection map is a specific type of mathematical function that takes points from one space and maps them onto another space, often simplifying the structure of the original space. This concept is crucial in understanding how certain spaces can be represented in lower dimensions, and it plays a key role in defining submersions and identifying regular values in differential topology.
Quotient Manifold: A quotient manifold is a type of manifold that is formed by taking a differentiable manifold and identifying points based on an equivalence relation, typically defined by a smooth action of a group. This process produces a new manifold where each point represents an entire equivalence class of points from the original manifold. Understanding quotient manifolds helps clarify the concepts of submersions and regular values, as they often arise in the study of how manifolds can be simplified or transformed through these identification processes.
Rank of a map: The rank of a map refers to the dimension of the image or the output space of a smooth map between manifolds. It indicates the maximum number of linearly independent vectors that can be obtained from the differential of the map at a given point. Understanding the rank is crucial when discussing concepts such as submersions and regular values, as it helps identify how well the map behaves locally around those points.
Regular Value: A regular value is a point in the target space of a smooth map such that the preimage of that point consists only of points where the differential of the map is surjective. This concept is important for understanding how smooth maps behave and has applications in various areas, including the implicit function theorem, submersions, and determining properties like the degree of a map.
Sard's Theorem: 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.
Smooth approximations: Smooth approximations refer to the methods used to create functions that closely mimic the properties of a given function while maintaining differentiability. This concept is particularly useful in contexts where one needs to work with functions that are not smooth or have discontinuities, allowing for the analysis of submersions and regular values. By constructing smooth approximations, one can simplify problems in differential topology and make it easier to apply various theorems related to differentiable maps.
Smooth manifold: A smooth manifold is a topological space that is locally similar to Euclidean space and has a globally defined differential structure, allowing for the smooth transition of functions. This concept is essential in many areas of mathematics and physics, as it provides a framework for analyzing shapes, curves, and surfaces with differentiable structures.
Submersion: Submersion is a type of smooth map between differentiable manifolds where the differential of the map is surjective at every point in its domain. This means that for each point in the target manifold, there are points in the source manifold that are mapped to it, allowing for a rich structure in differential topology and the exploration of properties like regular values and smoothness.
Surjective Differential: A surjective differential refers to a situation where the differential map of a smooth function between manifolds is onto, meaning it covers the entire tangent space at each point in the target manifold. This concept is critical because it ensures that the function achieves all possible directions in the tangent space, which has significant implications for the study of submersions and regular values. When a differential is surjective, it indicates that the function locally resembles a projection, providing useful insights into the topology and structure of the manifolds involved.
Tangent Bundle: The tangent bundle of a manifold is a new manifold that encapsulates all the tangent spaces of the original manifold at every point. It allows us to study how vectors can vary as we move around the manifold, creating a powerful framework for understanding concepts like differentiation, vector fields, and dynamics. The tangent bundle is fundamental in connecting ideas about tangent vectors and spaces to the behavior of smooth functions, and it plays a crucial role in applying partitions of unity and analyzing vector fields on manifolds.
Transition Functions: Transition functions are mathematical tools used in the study of manifolds that allow for the change of coordinates between overlapping charts. They facilitate the smooth transformation of one local representation of a manifold to another, ensuring that the manifold's structure is preserved under these transformations. These functions are crucial in establishing the compatibility of different coordinate systems, thus enabling a comprehensive understanding of the manifold's topology and geometry.
Vector Bundle: A vector bundle is a topological construct that consists of a base space and a family of vector spaces attached to each point in that space. It provides a way to study how vector spaces vary continuously over a manifold, allowing us to connect geometric concepts with algebraic structures. This concept is crucial when examining smooth maps, submersions, and regular values, as it allows for an understanding of how these mappings behave in a multi-dimensional context.