5.2 Immersions and their properties
2 min read•august 9, 2024
Immersions are smooth maps between that preserve local structure. They're crucial for understanding how lower-dimensional spaces can fit into higher-dimensional ones, even if they intersect themselves. This concept builds on our study of manifolds and their relationships.
Immersions allow us to explore complex geometric shapes and their properties. They help us visualize abstract mathematical concepts and provide a bridge between topology and differential geometry, key themes in our exploration of submanifolds and their mappings.
Immersions and Local Properties
Understanding Immersions and Their Characteristics
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- defines a between manifolds preserving local structure
- establishes a one-to-one correspondence between neighborhoods of points in the domain and range
- equals the dimension of the domain manifold at every point
- ensures the derivative map is one-to-one at each point
Mathematical Properties and Implications
- Immersions allow for local of manifolds into higher-dimensional spaces
- Local diffeomorphism property enables preservation of topological and differential structure in small neighborhoods
- Rank condition guarantees the map does not collapse dimensions locally
- Injective differential implies the tangent spaces are mapped isomorphically
Examples and Applications
- Immersion of a circle into a figure-eight shape in the plane (demonstrates )
- immersed in three-dimensional space (showcases a non-embeddable surface)
- Immersion of a sphere into a three-dimensional space with a single point of self-intersection (illustrates local injectivity)
- as an immersion from a surface to a sphere (applies in differential geometry)
Self-Intersections and Homotopy
Exploring Self-Intersections in Immersions
- Self-intersection occurs when distinct points in the domain map to the same point in the range
- Immersed submanifolds may have self-intersections while embedded submanifolds cannot
- Self-intersections can be transverse or tangential, affecting the local structure of the immersion
- Analysis of self-intersections provides insights into the global properties of the immersion
Immersion Theorem and Its Implications
- states conditions for the existence of immersions between manifolds
- Theorem relates the dimensions of the domain and codomain manifolds
- Provides a foundation for understanding when immersions are possible
- Applies to various geometric and topological problems involving manifolds
Regular Homotopy and Classification
- defines a continuous deformation between immersions
- Preserves the immersion property throughout the deformation
- Classifies immersions up to regular equivalence
- extends the concept to higher dimensions
Applications and Examples
- Immersion of a into three-dimensional space (demonstrates non-orientability)
- as an immersion of the projective plane in three-dimensional space (illustrates complex self-intersections)
- Regular homotopy between different immersions of a circle in the plane (shows topological equivalence)
- on the self-intersection number of immersed surfaces (connects to algebraic topology)
Key Terms to Review (27)
Boy's Surface: Boy's Surface is a non-orientable surface that cannot be embedded in three-dimensional Euclidean space without self-intersections. It is a classic example in differential topology and can be thought of as a model of a projective plane with a single cross-cap. This surface showcases important properties of immersions, particularly how they can exhibit unique characteristics like self-intersection and non-orientability.
Critical Point: A critical point is a point on a differentiable function where its derivative is either zero or undefined, indicating a potential local maximum, local minimum, or saddle point. Understanding critical points is essential in analyzing smooth maps, immersions, and their properties, as well as in applying various theorems related to differential topology.
Derivative of an Immersion: The derivative of an immersion is a mathematical concept that describes how a smooth map between manifolds behaves locally. Specifically, it refers to the pushforward of the tangent space at a point on the domain manifold, which is represented by a linear map that captures how vectors are transformed under the immersion. This concept is crucial in understanding how immersions preserve certain geometric structures and how they relate to other properties like injectivity and dimensionality.
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.
Differentiable structures: Differentiable structures refer to the ways in which a manifold can be given a smooth structure, allowing for the definition of differentiable functions and the application of calculus on it. This concept is crucial for understanding how manifolds behave and interact with differentiable maps, which are essential in analyzing their properties, especially when looking at immersions and the nuances of smoothness.
Embedding: An embedding is a type of function that allows one mathematical object to be treated as if it were contained within another, often preserving certain structures like topology and differentiability. This concept is crucial for understanding how submanifolds can be smoothly included in larger manifolds, impacting the way we analyze geometric and topological properties of spaces.
Folded Immersion: A folded immersion is a type of smooth immersion of a manifold where the image has self-intersections, but these intersections have a specific structure, allowing for some parts of the manifold to be locally folded over themselves. This concept ties into the study of immersions, focusing on how shapes can be manipulated and understood in terms of their differentiable properties, leading to fascinating implications in topology and geometry.
Gauss map: The Gauss map is a mathematical function that assigns to each point on a surface its corresponding normal vector, essentially mapping the surface into the unit sphere. This concept is crucial for understanding how surfaces behave and interact with their surrounding space, particularly in the context of immersions and computing the degree of specific maps.
Homotopy: Homotopy is a concept in topology that describes a continuous deformation between two continuous functions. It establishes a relation between two maps by allowing one to be transformed into the other through a series of intermediate steps, known as homotopies. This idea is crucial for understanding properties like when two paths can be continuously deformed into one another, which connects deeply with the study of immersions, transversality, degrees of maps, and fixed point theory.
Immersion: An immersion is a smooth map between differentiable manifolds that reflects the local structure of the manifolds, allowing for the differential structure to be preserved. This means that at each point in the domain, the map can be represented by a differentiable function whose derivative is injective, indicating that locally, the manifold can be thought of as being 'inserted' into another manifold without self-intersections.
Immersion of Surfaces: An immersion of surfaces refers to a smooth mapping from one manifold to another, where the differential of the mapping is injective at every point. This means that locally, the surface can be represented without self-intersections, allowing for a well-defined tangent space at each point. Immersions are crucial in understanding how surfaces can be placed in higher-dimensional spaces, affecting properties such as topology and geometry.
Immersion of the Circle: An immersion of the circle refers to a smooth function from the circle, typically denoted as $S^1$, into a Euclidean space, where the function is injective and its derivative is never zero at any point. This concept is essential in understanding how curves can be represented in higher-dimensional spaces while maintaining their distinctiveness and avoiding self-intersections. Immersions reveal the behavior of curves and are pivotal in studying topological properties and characteristics related to smooth mappings.
Immersion Theorem: The Immersion Theorem states that a smooth manifold can be immersed into a Euclidean space of sufficiently high dimension. This means that locally, around every point, the manifold can be represented as a subset of Euclidean space, preserving its differentiable structure. This theorem connects important concepts in differential topology, such as immersions and embeddings, helping to clarify how manifolds behave in higher dimensions.
Injective Differential: An injective differential is a linear map between tangent spaces that preserves the distinctness of vectors, meaning it maps distinct tangent vectors at a point to distinct vectors in the target space. This concept is essential for understanding immersions, as it ensures that the map locally resembles an embedding, allowing for the preservation of the manifold's structure. The injectiveness of the differential at a point indicates that the immersion is locally one-to-one around that point.
Klein Bottle: A Klein bottle is a non-orientable surface that cannot be embedded in three-dimensional Euclidean space without self-intersections. It is a one-sided surface, meaning if you travel along it, you can return to your starting point while being on what appears to be the 'other side.' This fascinating structure is crucial for understanding concepts related to immersions and the properties of manifolds, showcasing how surfaces can defy our typical intuitions about geometry and dimensions.
Local diffeomorphism: A local diffeomorphism is a smooth map between manifolds that has a smooth inverse in a neighborhood of every point in its domain. This means that near each point, the map behaves like a bijective function, allowing for smooth transitions between the two spaces. Local diffeomorphisms are important in understanding the structure of manifolds and play a crucial role in various mathematical concepts, including differentiable structures and smooth maps.
Manifolds: Manifolds are mathematical spaces that locally resemble Euclidean space and can be used to model complex shapes and structures. They are essential in differential topology as they provide the foundation for understanding curves, surfaces, and higher-dimensional spaces. Manifolds enable the study of properties that remain invariant under continuous transformations, making them crucial for analyzing immersions and understanding how various geometric structures evolve under transformations.
Möbius strip: A Möbius strip is a one-sided surface created by taking a rectangular strip of paper, giving it a half-twist, and then joining the ends together. This unique shape has fascinating properties, such as having only one boundary and being non-orientable, which means that if you travel along the surface, you can return to your starting point while being on the 'opposite' side. These features make the Möbius strip a classic example in the study of immersions and their properties.
Rank of an Immersion: The rank of an immersion refers to the maximum number of linearly independent tangent vectors at a point in the manifold where the immersion is defined. It plays a critical role in understanding how the immersion behaves locally and how it relates to the topology of the manifold. The rank helps determine whether the immersion can locally represent the manifold as a submanifold in a Euclidean space and provides insight into its geometric properties.
Regular Homotopy: Regular homotopy refers to a specific type of homotopy between two continuous maps (or immersions) from a manifold into another manifold, where the paths can be transformed into each other through a family of immersions that are smooth and maintain certain properties. This concept is crucial when analyzing how immersions can be continuously deformed while preserving their regularity. Regular homotopy highlights the distinction between different types of paths in topology, especially when considering the behavior and structure of curves and surfaces in higher-dimensional spaces.
Regular Immersion: A regular immersion is a smooth map between differentiable manifolds that is an immersion and has a regular value. This means that the differential of the map is injective at every point in the domain and the preimage of every regular value is a submanifold of the target manifold. Regular immersions help in understanding the structure of smooth manifolds and play a crucial role in various geometric properties.
Self-Intersection: Self-intersection refers to the phenomenon where a curve or surface intersects itself at one or more points. This concept is essential in the study of immersions, as it helps in understanding when a curve fails to be an immersion due to overlapping points in its image, affecting the overall properties and behavior of the mapping.
Smale-Hirsch Theory: The Smale-Hirsch Theory is a significant result in differential topology that deals with the classification of immersions of manifolds. It provides necessary and sufficient conditions for when one manifold can be immersed in another, specifically focusing on the relationship between the dimensions of the manifolds involved and the properties of their immersions. This theory connects deeply with concepts such as transversality, regularity of immersions, and how these interactions can affect the topological structure.
Smooth map: A smooth map is a function between differentiable manifolds that has continuous derivatives of all orders. This concept is essential in understanding the behavior of maps in differential topology, as smooth maps preserve the structure of manifolds and allow for analysis using calculus tools. The properties of smooth maps play a critical role in determining the characteristics of immersions, analyzing the differential of a map, exploring transversality, and defining degrees of mappings.
Tangent Vector: A tangent vector is a mathematical object that represents a direction and rate of change at a specific point on a curve or manifold. It provides a way to understand how a function behaves locally around that point, which is crucial in analyzing properties like differentiability and smoothness in various contexts. Tangent vectors also form the foundation for defining tangent spaces, which allow for the generalization of concepts from calculus to more complex geometric structures.
Transversality Theorem: The Transversality Theorem is a fundamental result in differential topology that describes how submanifolds intersect each other in a smooth manifold. Specifically, it provides conditions under which the intersection of two submanifolds is 'transverse,' meaning their tangent spaces at each intersection point together span the tangent space of the ambient manifold. This concept is crucial for understanding immersions and embeddings, as well as establishing properties like the existence of regular values in smooth mappings.
Whitney's Theorem: Whitney's Theorem states that for a smooth manifold, the set of immersions into Euclidean space can be characterized by certain properties related to the dimensions of the manifold and the ambient space. It highlights the relationship between immersions, transversality, and the critical values of smooth functions, establishing key connections in differential topology that are essential for understanding the behavior of mappings between manifolds.