Embedded and immersed submanifolds are key concepts in differential geometry. They describe how one manifold can be contained within another of higher dimension, with embeddings being more restrictive and immersions allowing for self-intersections.
These concepts are crucial for understanding the geometry of curves and surfaces in higher-dimensional spaces. They provide tools for analyzing constraints in mechanics, modeling spacetime in physics, and creating complex shapes in computer graphics.
Definition of embedded submanifolds
Embedded submanifolds are a fundamental concept in differential geometry that describe how a manifold can be contained within another manifold of higher dimension
Embeddedness is a stronger condition than immersion, requiring the submanifold to inherit its topology and smooth structure from the ambient manifold
Topological embedding
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A topological embedding is a from a topological space M to a subspace of another topological space N
The image of M under the embedding is an of N
The embedding preserves the topological properties of M, such as continuity and convergence of sequences
Smooth embedding
A smooth embedding is a f:M→N between smooth manifolds that is also a topological embedding
The differential dfp:TpM→Tf(p)N is injective at every point p∈M
Smooth embeddings preserve the smooth structure of M, allowing calculus and differential geometry techniques to be applied to the submanifold
Codimension of embedded submanifolds
The codimension of an embedded submanifold M in an ambient manifold N is the difference between their dimensions: codim(M)=dim(N)−dim(M)
Codimension measures the number of dimensions "lost" when embedding M into N
Submanifolds of codimension 1 are called hypersurfaces (curves in surfaces, surfaces in 3-manifolds)
Properties of embedded submanifolds
Embedded submanifolds inherit various properties from their ambient manifolds, which simplifies their study and allows for the application of results from the ambient space
The induced structures on embedded submanifolds are compatible with those of the ambient manifold, ensuring consistency in calculations and theorems
Induced topology from ambient manifold
The subspace topology on an embedded submanifold M⊂N is induced by the topology of the ambient manifold N
Open sets in M are precisely the intersections of open sets in N with M
The induced topology makes the inclusion map i:M→N continuous
Induced smooth structure
The smooth structure on an embedded submanifold M⊂N is induced by the smooth structure of the ambient manifold N
Smooth functions on M are precisely the restrictions of smooth functions on N to M
The induced smooth structure makes the inclusion map i:M→N smooth
Tangent spaces of embedded submanifolds
The TpM at a point p of an embedded submanifold M⊂N is a linear subspace of the tangent space TpN of the ambient manifold
TpM consists of the tangent vectors to curves in M passing through p
The differential of the inclusion map dip:TpM→TpN is an injective linear map
Examples of embedded submanifolds
Concrete examples help to illustrate the abstract concepts of embedded submanifolds and their properties
Studying these examples provides insight into the geometry and topology of embedded submanifolds in various contexts
Spheres in Euclidean space
The n-sphere Sn is an embedded submanifold of the Euclidean space Rn+1
Sn={x∈Rn+1:∥x∥=1} is the set of points at unit distance from the origin
The inclusion map i:Sn→Rn+1 is a smooth embedding of codimension 1
Graphs of smooth functions
The graph of a smooth function f:M→N between manifolds is an embedded submanifold of the product manifold M×N
Graph(f)={(x,f(x)):x∈M} is the set of points (x,y) with y=f(x)
The graph embedding Γf:M→M×N,x↦(x,f(x)) is a smooth embedding of codimension dim(N)
Level sets of smooth maps
The level set of a smooth map f:M→N at a regular value y∈N is an embedded submanifold of M
f−1(y)={x∈M:f(x)=y} is the set of points in M mapped to y by f
The codimension of the level set is equal to the dimension of N, and the inclusion map i:f−1(y)→M is a smooth embedding
Definition of immersed submanifolds
Immersed submanifolds are a generalization of embedded submanifolds that allow for self-intersections and non-injective maps
Immersions capture the local geometry of a manifold while allowing for more flexible global behavior
Immersion as a smooth map
An immersion is a smooth map f:M→N between manifolds whose differential dfp:TpM→Tf(p)N is injective at every point p∈M
Immersions preserve the local geometry of M, mapping tangent spaces injectively
Unlike embeddings, immersions need not be injective globally, allowing for self-intersections
Local embedding property
Every immersion f:M→N is locally an embedding
For each point p∈M, there exists a neighborhood U of p such that the restriction f∣U:U→N is an embedding
Immersions are locally indistinguishable from embeddings, but their global structure may differ
Codimension of immersed submanifolds
The codimension of an M in an ambient manifold N is the difference between their dimensions: codim(M)=dim(N)−dim(M)
As with embedded submanifolds, codimension measures the number of dimensions "lost" when immersing M into N
Immersed submanifolds of codimension 0 are local diffeomorphisms (open maps)
Properties of immersed submanifolds
Immersed submanifolds share some properties with embedded submanifolds but may exhibit different global behavior
Understanding the properties of immersed submanifolds is crucial for studying their geometry and topology
Induced topology vs ambient topology
Unlike embedded submanifolds, immersed submanifolds may not inherit their topology from the ambient manifold
The induced topology on an immersed submanifold M is generally finer than the subspace topology from the ambient manifold N
Continuous functions on M with respect to the induced topology may not extend continuously to N
Tangent spaces of immersed submanifolds
The tangent space TpM at a point p of an immersed submanifold M⊂N is a linear subspace of the tangent space Tf(p)N of the ambient manifold
TpM consists of the tangent vectors to curves in M passing through p, mapped injectively by the differential dfp
The tangent spaces of an immersed submanifold may intersect non-trivially at self-intersection points
Self-intersections in immersed submanifolds
Immersed submanifolds may exhibit self-intersections, where distinct points in the domain manifold are mapped to the same point in the ambient manifold
Self-intersections occur when the immersion f:M→N is not injective
The local geometry of an immersed submanifold at a self-intersection point is determined by the tangent spaces of the intersecting branches
Examples of immersed submanifolds
Studying concrete examples of immersed submanifolds helps to develop intuition for their properties and behavior
These examples showcase the flexibility and variety of immersed submanifolds in different settings
Lemniscate curve in the plane
The lemniscate of Bernoulli is an immersed curve in the plane R2 defined by the equation (x2+y2)2=a2(x2−y2)
The lemniscate has a self-intersection at the origin and is symmetric about both the x and y axes
The immersion f:R→R2 given by f(t)=(acost/(1+sin2t),asintcost/(1+sin2t)) parametrizes the lemniscate
Klein bottle in 4-dimensional space
The Klein bottle is a non-orientable surface that can be immersed in 4-dimensional Euclidean space R4
It is obtained by gluing the edges of a rectangle with a twist, resulting in a single self-intersection
The Klein bottle cannot be embedded in R3, but it admits an immersion in R4 given by f(u,v)=((r+cosu/2)cosv,(r+cosu/2)sinv,sinu/2cosv/2,sinu/2sinv/2)
Immersions of the circle in the plane
The circle S1 admits various immersions into the plane R2, showcasing different self-intersection patterns
The figure-eight immersion f:S1→R2 given by f(t)=(sint,sin2t) has a single self-intersection at the origin
The immersion f(t)=(cos3t,sin3t) creates a trifolium with three self-intersections, resembling a three-leaf clover
Comparison of embedded vs immersed submanifolds
Understanding the differences between embedded and immersed submanifolds is essential for choosing the appropriate concept for a given problem
The comparison highlights the trade-offs between the stricter conditions of embeddedness and the flexibility of immersions
Topological differences
Embedded submanifolds inherit their topology from the ambient manifold, while immersed submanifolds may have a finer topology
Embedded submanifolds are homeomorphic to their image in the ambient manifold, whereas immersed submanifolds may not be
Continuous functions on embedded submanifolds extend continuously to the ambient manifold, which may not be true for immersed submanifolds
Smooth structure differences
Both embedded and immersed submanifolds inherit their smooth structure from the ambient manifold
Smooth functions on embedded submanifolds are restrictions of smooth functions on the ambient manifold
Smooth functions on immersed submanifolds may not extend smoothly to the ambient manifold due to self-intersections
Tangent space properties
Tangent spaces of embedded submanifolds are linear subspaces of the tangent spaces of the ambient manifold
Tangent spaces of immersed submanifolds are mapped injectively into the tangent spaces of the ambient manifold
At self-intersection points of immersed submanifolds, the tangent spaces of the intersecting branches may intersect non-trivially
Applications of embedded and immersed submanifolds
Embedded and immersed submanifolds have numerous applications in various fields, demonstrating their practical importance
These applications rely on the geometric and topological properties of submanifolds to model and analyze complex systems
Constraint systems in mechanics
Embedded submanifolds can be used to model constraint systems in classical mechanics
The configuration space of a constrained system is an embedded submanifold of the unconstrained configuration space
Lagrangian and Hamiltonian mechanics can be formulated on embedded submanifolds, taking the constraints into account
Submanifold geometry in physics
Submanifold geometry plays a crucial role in various areas of physics, such as general relativity and string theory
In general relativity, spacetime is modeled as a 4-dimensional Lorentzian manifold, and physical objects are represented by embedded submanifolds (worldlines, worldsheets)
String theory describes fundamental particles as vibrating strings, which can be viewed as embedded or immersed submanifolds of a higher-dimensional spacetime
Modeling surfaces in computer graphics
Immersed submanifolds, particularly surfaces, are widely used in computer graphics for modeling and rendering 3D objects
Parametric surfaces, such as Bézier surfaces and NURBS, are immersions of rectangular domains into 3D space
Subdivision surfaces, obtained by recursively refining a coarse mesh, provide a flexible way to model complex shapes with self-intersections and non-manifold features
Key Terms to Review (18)
Circle in R^2: A circle in R^2 is a set of points in a two-dimensional Euclidean space that are all equidistant from a fixed point, known as the center. The distance from the center to any point on the circle is called the radius, and this can be expressed mathematically as the set of points satisfying the equation $$x^2 + y^2 = r^2$$ for a circle centered at the origin, where 'r' is the radius. This geometric shape serves as a fundamental example of a 1-dimensional embedded submanifold in a higher-dimensional space.
Differentiable manifold: A differentiable manifold is a topological space that is locally similar to Euclidean space and allows for the definition of smooth functions. This structure enables calculus to be performed on the manifold, facilitating the study of its geometric and topological properties. Differentiable manifolds serve as a foundation for various mathematical concepts, including smooth functions, embedded or immersed submanifolds, and Morse theory, which all explore different aspects of these smooth structures.
Embedded submanifold: An embedded submanifold is a subset of a manifold that is itself a manifold and is smoothly placed within the larger manifold in such a way that the inclusion map is an embedding. This means that the submanifold retains its own topological and differentiable structure while fitting neatly into the surrounding manifold. Embedded submanifolds can be thought of as lower-dimensional surfaces or shapes within higher-dimensional spaces, preserving properties like continuity and differentiability.
Homeomorphism: A homeomorphism is a continuous function between two topological spaces that has a continuous inverse, essentially providing a way to show that two spaces are 'the same' in a topological sense. This concept plays a crucial role in understanding the properties of spaces, as it indicates that these spaces can be transformed into one another without tearing or gluing, preserving their topological characteristics. The significance of homeomorphisms can be seen in various mathematical contexts, from defining equivalence classes of shapes to analyzing structures like manifolds and submanifolds.
Homotopy: Homotopy is a concept in topology that refers to the idea of continuously transforming one function into another within a certain space. This notion allows us to classify functions based on their ability to be deformed into each other, which plays a crucial role in understanding the properties of spaces and the relationships between different shapes. In the context of embedded and immersed submanifolds, it helps in studying how these submanifolds can be transformed, while in the context of foliations, it aids in understanding the structure of leaves and how they relate through continuous deformations. Furthermore, in comparison geometry and Toponogov's theorem, homotopy can provide insights into the geometric properties of manifolds by comparing them under continuous transformations.
Immersed submanifold: An immersed submanifold is a subset of a manifold that has the structure of a manifold itself and is smoothly embedded in a larger manifold, but may not be embedded in a way that maintains its topology. This means that locally, it behaves like a manifold, but globally, it can intersect itself or have self-intersections, which is what differentiates it from an embedded submanifold.
Immersion Theorem: The immersion theorem states that if a manifold can be represented as an immersion into a Euclidean space, then it has a certain level of local structure that allows for a smooth, differentiable relationship with the surrounding space. This concept is crucial when discussing embedded and immersed submanifolds, as it provides insights into how these structures can be understood and analyzed in higher dimensions.
John Nash: John Nash was a prominent mathematician known for his groundbreaking contributions to game theory, which is crucial for understanding strategies in competitive situations. His work laid the foundation for various concepts, including Nash equilibrium, which describes a stable state in a game where no player can benefit by changing their strategy unilaterally. His insights have far-reaching implications in economics, political science, and even biology, connecting deeply with concepts such as optimization and decision-making strategies.
Michael Spivak: Michael Spivak is a prominent mathematician known for his contributions to differential geometry and for authoring influential texts that bridge mathematics and physics. His work, particularly in the realm of manifold theory, has provided essential insights into the concepts of embedded and immersed submanifolds, helping to shape modern understanding in these areas.
Normal Bundle: The normal bundle of a submanifold is a vector bundle that captures the directions in which the submanifold can be displaced within its ambient manifold. It consists of all the vectors that are orthogonal to the tangent space of the submanifold at each point, providing essential information about the geometry of both the submanifold and the surrounding manifold. This concept is crucial for understanding how submanifolds interact with their environment, including properties like curvature and metrics.
Projection Map: A projection map is a mathematical function that maps points from one space onto another, typically used to describe how a manifold can be represented in a lower-dimensional space. This concept is essential when considering embedded and immersed submanifolds, as it allows for the understanding of how these submanifolds relate to their ambient spaces and how they can be visualized or analyzed within a more manageable framework.
Riemannian manifold: A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which allows for the measurement of lengths of curves, angles between vectors, and areas of surfaces. This structure provides the geometric framework needed to study concepts like curvature and distance in a way that generalizes the familiar properties of Euclidean space.
Smooth Map: A smooth map is a function between two smooth manifolds that preserves the structure of the manifolds by ensuring that the map is infinitely differentiable. This means that when you take derivatives of the map, all of them exist, making it crucial in studying the relationships and properties of smooth manifolds. Smooth maps are foundational in understanding how different geometric structures interact and play a vital role in various advanced topics like submersions and embeddings.
Submersion: Submersion is a smooth and surjective differential map between differentiable manifolds, where the differential at each point is surjective. This concept is vital in understanding how one manifold can be 'mapped down' onto another, preserving certain geometric structures. Submersions are particularly important in the context of studying Riemannian submersions and the behavior of embedded and immersed submanifolds, as they provide insights into how different geometric properties interact when transitioning from one manifold to another.
Tangent Space: The tangent space at a point on a manifold is a vector space that consists of all possible directions in which one can tangentially pass through that point. This concept allows us to generalize the notion of derivatives from calculus to the context of manifolds, enabling the study of how functions behave locally around points on these complex structures.
Topological Manifold: A topological manifold is a topological space that locally resembles Euclidean space and is equipped with a topology that allows for the definition of continuous functions. This means that for every point in the manifold, there exists a neighborhood that can be mapped homeomorphically to an open subset of Euclidean space. Topological manifolds serve as the foundational concept for more advanced structures, such as smooth manifolds and embedded submanifolds, which require additional structure like differentiability or immersion.
Torus in R^3: A torus in R^3 is a doughnut-shaped surface generated by revolving a circle around an axis that does not intersect the circle. This shape is characterized by its two distinct radii: the major radius (distance from the center of the tube to the center of the torus) and the minor radius (radius of the tube itself). The torus serves as an essential example of a two-dimensional embedded submanifold in three-dimensional Euclidean space.
Whitney Embedding Theorem: The Whitney Embedding Theorem states that every smooth manifold can be embedded into Euclidean space of sufficiently high dimension. This theorem is significant because it provides a way to visualize and analyze manifolds using familiar geometric concepts, allowing for the study of their properties in a more tangible context.