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.
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A regular submanifold must be locally diffeomorphic to an open subset of Euclidean space, maintaining compatibility with the manifold structure.
Regular submanifolds can be defined using level sets of smooth functions, where the regularity condition ensures they behave well under differentiable maps.
The dimension of a regular submanifold is always less than that of the ambient manifold, indicating its relative 'size' within that space.
Examples of regular submanifolds include curves and surfaces within higher-dimensional spaces, such as circles or spheres embedded in Euclidean space.
The tangent space of a regular submanifold at any point can be identified with a subset of the tangent space of the ambient manifold at the same point.
Review Questions
How does the concept of an embedding relate to the definition of a regular submanifold?
An embedding is crucial to understanding regular submanifolds because it ensures that the inclusion map from the submanifold to the ambient manifold is smooth and preserves the manifold structure. This means that when you look at the submanifold, it behaves like its own manifold while being smoothly integrated into the larger context. In this way, embeddings allow for both local and global properties of the manifolds to be maintained.
Discuss how level sets can be used to construct examples of regular submanifolds.
Level sets are valuable for constructing regular submanifolds since they represent the pre-image of constant values under smooth functions defined on a manifold. To ensure that these level sets are indeed regular submanifolds, one must check that they meet the regularity conditions, typically using the implicit function theorem. If a level set is defined as the zero set of a smooth function whose differential is non-zero at points in question, then it forms a regular submanifold.
Evaluate the implications of defining a regular submanifold in terms of its tangent space properties within an ambient manifold.
Defining a regular submanifold in terms of its tangent space allows us to make deeper connections between local geometry and global topology. Specifically, each point on a regular submanifold has a tangent space that aligns with part of the ambient manifold's tangent space. This alignment reveals how smoothly one can transition between points on the submanifold and those on the ambient manifold, highlighting their intrinsic relationship. Moreover, examining these tangent spaces aids in understanding differentiable structures and geometric behavior at local points.
An embedding is a smooth map between manifolds that is both injective and an immersion, meaning it preserves the manifold's structure in a way that allows for local homeomorphism.
An immersion is a differentiable function between manifolds whose differential is injective at every point, allowing for a smooth, curved mapping into a higher-dimensional space.
The tangent space at a point on a manifold consists of all possible directions in which one can tangentially pass through that point, providing a vector space that captures local geometric properties.