8.1 Embedded and immersed submanifolds

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

• A topological embedding is a homeomorphism from a topological space $M$ to a subspace of another topological space $N$
• The image of $M$ under the embedding is an embedded submanifold of $N$
• The embedding preserves the topological properties of $M$, such as continuity and convergence of sequences

Smooth embedding

• A smooth embedding is a smooth map $f: M \to N$ between smooth manifolds that is also a topological embedding
• The differential $df_p: T_pM \to T_{f(p)}N$ is injective at every point $p \in M$
• 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: $\text{codim}(M) = \dim(N) - \dim(M)$
• 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 $M \subset N$ 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 $i: M \to N$ continuous

Induced smooth structure

• The smooth structure on an embedded submanifold $M \subset N$ 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 $i: M \to N$ smooth

Tangent spaces of embedded submanifolds

• The tangent space $T_pM$ at a point $p$ of an embedded submanifold $M \subset N$ is a linear subspace of the tangent space $T_pN$ of the ambient manifold
• $T_pM$ consists of the tangent vectors to curves in $M$ passing through $p$
• The differential of the inclusion map $di_p: T_pM \to T_pN$ 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 $S^n$ is an embedded submanifold of the Euclidean space $\mathbb{R}^{n+1}$
• $S^n = \{x \in \mathbb{R}^{n+1} : \|x\| = 1\}$ is the set of points at unit distance from the origin
• The inclusion map $i: S^n \to \mathbb{R}^{n+1}$ is a smooth embedding of codimension 1

Graphs of smooth functions

• The graph of a smooth function $f: M \to N$ between manifolds is an embedded submanifold of the product manifold $M \times N$
• $\text{Graph}(f) = \{(x, f(x)) : x \in M\}$ is the set of points $(x, y)$ with $y = f(x)$
• The graph embedding $\Gamma_f: M \to M \times N, x \mapsto (x, f(x))$ is a smooth embedding of codimension $\dim(N)$

Level sets of smooth maps

• The level set of a smooth map $f: M \to N$ at a regular value $y \in N$ is an embedded submanifold of $M$
• $f^{-1}(y) = \{x \in M : f(x) = y\}$ 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 $i: f^{-1}(y) \to M$ 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 $f: M \to N$ between manifolds whose differential $df_p: T_pM \to T_{f(p)}N$ is injective at every point $p \in M$
• 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 $f: M \to N$ is locally an embedding
• For each point $p \in M$, there exists a neighborhood $U$ of $p$ such that the restriction $f|_U: U \to N$ is an embedding
• Immersions are locally indistinguishable from embeddings, but their global structure may differ

Codimension of immersed submanifolds

• The codimension of an immersed submanifold $M$ in an ambient manifold $N$ is the difference between their dimensions: $\text{codim}(M) = \dim(N) - \dim(M)$
• 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 $T_pM$ at a point $p$ of an immersed submanifold $M \subset N$ is a linear subspace of the tangent space $T_{f(p)}N$ of the ambient manifold
• $T_pM$ consists of the tangent vectors to curves in $M$ passing through $p$, mapped injectively by the differential $df_p$
• 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 $f: M \to N$ 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 $\mathbb{R}^2$ defined by the equation $(x^2 + y^2)^2 = a^2(x^2 - y^2)$
• The lemniscate has a self-intersection at the origin and is symmetric about both the $x$ and $y$ axes
• The immersion $f: \mathbb{R} \to \mathbb{R}^2$ given by $f(t) = (a \cos t / (1 + \sin^2 t), a \sin t \cos t / (1 + \sin^2 t))$ 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 $\mathbb{R}^4$
• 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 $\mathbb{R}^3$, but it admits an immersion in $\mathbb{R}^4$ given by $f(u, v) = ((r + \cos u/2)\cos v, (r + \cos u/2)\sin v, \sin u/2 \cos v/2, \sin u/2 \sin v/2)$

Immersions of the circle in the plane

• The circle $S^1$ admits various immersions into the plane $\mathbb{R}^2$, showcasing different self-intersection patterns
• The figure-eight immersion $f: S^1 \to \mathbb{R}^2$ given by $f(t) = (\sin t, \sin 2t)$ has a single self-intersection at the origin
• The immersion $f(t) = (\cos^3 t, \sin^3 t)$ 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