5.1 Submanifolds and embeddings
3 min read•august 9, 2024
Submanifolds and embeddings are key concepts in differential topology. They allow us to study how smaller spaces fit inside larger ones, preserving important geometric and topological properties. This topic builds on our understanding of manifolds and maps between them.
These ideas are crucial for analyzing complex shapes and spaces. By looking at submanifolds and embeddings, we can break down tricky problems into more manageable pieces and gain insights into the structure of geometric objects.
Submanifolds and Embeddings
Defining Submanifolds and Embeddings
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- represents a subset of a manifold that inherits its own manifold structure
- Consists of a subset S of a manifold M with a topology and induced from M
- Must satisfy certain smoothness and dimensionality conditions
- describes a smooth injective immersion with additional properties
- Maps one manifold into another while preserving its topological structure
- Requires the map to be a homeomorphism onto its image
- provides a natural way to view a submanifold within its ambient space
- Defined as i: S → M, where S is a submanifold of M
- Preserves the structure of S within the larger manifold M
Types of Submanifolds
- exhibits particularly nice local properties
- Can be locally described by equations in a coordinate chart
- Allows for simpler analysis and manipulation in many cases
- generalizes the concept of a regular submanifold
- Allows for self-intersections and more complex global behavior
- Defined using an immersion rather than an embedding
- combines properties of regular and immersed submanifolds
- Requires the submanifold to be both immersed and a topological subspace
- Ensures a well-behaved global structure without self-intersections
Codimension and Tangent Spaces
Understanding Codimension
- measures the difference in dimensions between a manifold and its submanifold
- Calculated as codim(S) = dim(M) - dim(S), where S is a submanifold of M
- Provides insight into the relative "size" of the submanifold within its ambient space
- Codimension 1 submanifolds hold special significance in many applications
- Include hypersurfaces in higher-dimensional spaces (surfaces in 3D space)
- Often arise as level sets of smooth functions
Exploring Tangent Spaces and Normal Bundles
- represents the set of all tangent vectors at a point on a manifold
- Denoted as T_pM for a point p on manifold M
- Forms a vector space with the same as the manifold
- For a submanifold S of M, the tangent space T_pS can be viewed as a subspace of T_pM
- Allows for the study of how S is "embedded" in M at each point
- Crucial for understanding the local geometry of the submanifold
- encompasses all vectors perpendicular to a submanifold at each point
- Defined as the orthogonal complement of T_pS in T_pM
- Dimension of the normal bundle equals the codimension of S in M
- Plays a key role in studying the relationship between a submanifold and its ambient space
Whitney Embedding Theorem
Fundamentals of the Whitney Embedding Theorem
- provides a powerful result about embedding manifolds in Euclidean spaces
- States that any smooth n-dimensional manifold can be smoothly embedded in
- Guarantees the existence of an embedding, not its explicit construction
- Theorem applies to both compact and non-compact manifolds
- For compact manifolds, the embedding dimension can be reduced to
- Non-compact manifolds may require the full dimensions
- Implications extend beyond pure mathematics to areas like data visualization and dimensionality reduction
- Suggests that high-dimensional data can often be meaningfully represented in lower dimensions
- Provides theoretical justification for techniques like manifold learning in machine learning
Applications and Extensions
- Whitney's theorem serves as a foundation for more refined embedding results
- Leads to tighter bounds for specific classes of manifolds (surfaces can be embedded in )
- Inspires research into optimal embedding dimensions for various manifold types
- Practical applications arise in computer graphics and scientific visualization
- Enables representation of complex geometric objects in manageable dimensions
- Facilitates analysis and manipulation of high-dimensional data sets
- Connections to other areas of mathematics include:
- Differential topology (study of smooth structures on manifolds)
- (relating topological properties to algebraic invariants)
- Geometric measure theory (analyzing measures on submanifolds of Euclidean space)
Key Terms to Review (19)
Algebraic Topology: Algebraic topology is a branch of mathematics that uses algebraic methods to study topological spaces and their properties. By associating algebraic structures, like groups, to these spaces, it allows for a deeper understanding of their shape and connectivity. This approach is crucial for analyzing submanifolds and embeddings as well as understanding the implications of the Whitney Embedding Theorem.
Boundary: A boundary is a fundamental concept in topology that refers to the dividing line or surface that separates a space from its exterior. In various contexts, boundaries help define the limits of a space, whether in terms of submanifolds within manifolds, properties of topological spaces, or when dealing with integrals of differential forms on manifolds. Understanding boundaries is crucial for analyzing the structure and behavior of mathematical objects.
Circle as a Submanifold of the Plane: A circle can be defined as a submanifold of the plane by considering it as a one-dimensional manifold that is embedded in two-dimensional Euclidean space. This means that the circle has a smooth structure and locally resembles a line while being situated within a higher-dimensional space, allowing us to study its properties through the lens of differential topology.
Codimension: Codimension is a concept in topology that measures the difference between the dimensions of a manifold and its submanifold. It quantifies how many dimensions are 'lost' when considering a submanifold within a larger manifold, indicating the extent to which the submanifold is embedded within the larger space. This idea plays a crucial role in understanding how submanifolds relate to their ambient manifolds, particularly in terms of embeddings, examples of manifolds, and transversality properties.
Differentiable Structure: A differentiable structure on a manifold is a way of defining how to differentiate functions on that manifold, allowing us to consider smooth functions and smooth transitions between charts. This structure is crucial because it enables the application of calculus in more abstract settings, which can then be connected to important concepts like submanifolds, examples of manifolds, partitions of unity, and embedding theorems.
Differential Geometry: Differential geometry is a field of mathematics that uses techniques of calculus and algebra to study the properties and structures of geometric objects, particularly those that can be described by smooth curves and surfaces. It bridges the gap between algebraic geometry and classical geometry, allowing for the analysis of shapes through their curvature, connections, and other intrinsic features. This area of study is crucial when looking at submanifolds and embeddings, as it helps describe how these smaller spaces fit into larger geometric frameworks.
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.
Embedded Submanifold: An embedded submanifold is a subset of a manifold that inherits its manifold structure through a smooth embedding, meaning it can be treated as a manifold in its own right while being contained within a larger manifold. This concept plays a crucial role in understanding the relationship between different manifolds, particularly in how they can share geometric and topological properties while maintaining distinct identities.
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.
Immersed submanifold: An immersed submanifold is a subset of a manifold that has a differentiable structure and can be locally represented as a differentiable map from a Euclidean space into the larger manifold. It retains some properties of submanifolds, but unlike embedded submanifolds, it may self-intersect and does not necessarily have a topology that matches the ambient manifold in every point. This concept connects with smooth maps, which describe how these structures can be smoothly related to one another.
Inclusion Map: An inclusion map is a type of function that takes elements from a subset and maps them into a larger set, typically seen in the context of topology and differential geometry. It captures the idea of treating a smaller space as part of a larger space, preserving the structure of the original subset while allowing it to be analyzed in a broader setting. This concept is crucial when discussing submanifolds and embeddings, where understanding how these smaller spaces fit within larger manifolds is fundamental.
Normal Bundle: A normal bundle is a vector bundle associated with an embedding of a manifold into another manifold, capturing how the embedded manifold sits within the larger space. It consists of all the vectors that are perpendicular to the tangent space of the embedded manifold at each point, essentially measuring the 'thickness' or 'direction' away from the embedded manifold. This concept helps in understanding properties like curvature and geometric structures of submanifolds as well as their implications in broader mathematical contexts.
Regular Submanifold: A regular submanifold is a subset of a manifold that is itself a manifold, with the inclusion map being an embedding. This means it has a well-defined dimension and structure that can be smoothly related to the larger manifold. Regular submanifolds retain the topological and differentiable properties of the ambient manifold, allowing for smooth transitions and operations between the two.
Smooth embedding: A smooth embedding is a type of function that maps one manifold into another in a way that preserves the smooth structure. This means that the function is not only continuous but also has continuous derivatives up to any desired order, making it a very 'nice' way to embed one space into another. Smooth embeddings are important because they allow us to study the properties of submanifolds within larger manifolds, maintaining essential geometric and topological features.
Sphere as a submanifold of Euclidean space: A sphere as a submanifold of Euclidean space is a set of points in a higher-dimensional space that are all equidistant from a fixed point, referred to as the center. It serves as an example of a smooth, compact manifold that can be embedded in higher dimensions, allowing for the exploration of geometric and topological properties. The sphere can be understood through its relationship with embeddings and submanifolds, showcasing how lower-dimensional structures fit within higher-dimensional spaces.
Submanifold: A submanifold is a subset of a manifold that is itself a manifold, typically characterized by being defined as the zero set of a smooth map or as an embedded subset. Submanifolds maintain the smooth structure of the ambient manifold, which allows for important geometric and topological properties to be preserved. They play crucial roles in various mathematical concepts, including differential maps and transversality, helping to understand the interactions between different manifolds.
Tangent Space: The tangent space at a point on a manifold is a vector space that consists of all possible tangent vectors at that point, representing the directions in which one can tangentially pass through the point. This concept is crucial for understanding how manifolds behave locally and connects to various mathematical ideas like differentiability, embeddings, and smooth structures.
Topological embedding: A topological embedding is a function between two topological spaces that preserves the structure of the spaces and allows for one space to be represented within another without any distortion. This means that the embedded space retains its properties, like open sets, and behaves like a subspace of the larger space, which is crucial for understanding the relationships between different geometric shapes and manifolds.
Whitney Embedding Theorem: The Whitney Embedding Theorem states that any smooth manifold can be embedded as a smooth submanifold of Euclidean space. This result is fundamental in differential topology because it shows how abstract manifolds can be realized in a more familiar geometric setting, allowing for the application of Euclidean tools to study their properties.