2.2 Smooth maps and their properties
3 min read•august 9, 2024
Smooth maps are the backbone of differential topology, allowing us to study relationships between manifolds. They're like the cool kids of functions, with continuous derivatives of all orders, making them super predictable and easy to work with.
These maps have awesome properties, like preserving topological features and playing nice when combined. Diffeomorphisms take it up a notch, giving us a way to say when two manifolds are essentially the same, just bent or stretched differently.
Smooth Maps and Functions
Differentiability and Smoothness
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- describes a function between manifolds with continuous derivatives of all orders
- refers to a function with k continuous derivatives
- C^1 functions have continuous first derivatives
- C^2 functions have continuous first and second derivatives
- denotes a smooth function with continuous derivatives of all orders
- Also known as infinitely differentiable functions
- Smoothness ensures predictable behavior and allows for advanced mathematical analysis
- Applications include modeling physical phenomena (fluid dynamics, electromagnetic fields)
Properties of Smooth Functions
- results in a smooth function
- Smooth functions preserve topological properties of manifolds
- applies to smooth functions with non-zero Jacobian determinant
- Taylor series expansion can approximate smooth functions around a point
- Smooth functions form a vector space under addition and scalar multiplication
Diffeomorphisms
Definition and Characteristics
- represents a bijective smooth map with a smooth inverse
- Preserves the differential structure between manifolds
- Establishes an equivalence relation between smooth manifolds
- refers to a smooth map that is locally invertible
- Behaves like a diffeomorphism in a small neighborhood of each point
- Diffeomorphisms form a group under composition ()
Applications and Examples
- Used to study symmetries in physics (Lie groups)
- Coordinate transformations in differential geometry often involve diffeomorphisms
- Möbius transformations on the complex plane (conformal maps)
- Time-evolution maps in dynamical systems can be diffeomorphisms
- Stereographic projection maps a sphere to a plane (except for one point)
Immersions and Submersions
Immersions and Their Properties
- describes a smooth map with injective differential at every point
- Locally injects the domain manifold into the target manifold
- Immersed submanifolds may have self-intersections (figure-eight curve)
- guarantees immersions of n-manifolds into R^(2n)
- Immersions preserve local geometric properties (tangent spaces, curvature)
Submersions and Embeddings
- represents a smooth map with surjective differential at every point
- Locally projects the domain manifold onto the target manifold
- Fibers of a submersion form submanifolds of the domain
- combines properties of an immersion with a homeomorphism onto its image
- Produces a submanifold without self-intersections
- ensures embeddings of n-manifolds into R^(2n+1)
Critical Points and Values
Critical Point Analysis
- denotes a point in the target manifold with no critical points in its preimage
- occurs where the differential of a smooth map is not surjective
- Differential has rank less than the dimension of the target manifold
- studies the relationship between critical points and manifold topology
- measures the number of negative eigenvalues of the Hessian matrix
- states that the set of critical values has measure zero
Applications in Optimization and Physics
- Critical points play a crucial role in optimization problems (local extrema)
- Lagrange multipliers method uses critical points to find constrained extrema
- Catastrophe theory analyzes sudden changes in behavior near degenerate critical points
- Hamiltonian systems in classical mechanics utilize critical points of the Hamiltonian function
- Morse-Smale complexes decompose manifolds based on gradient flow between critical points
Key Terms to Review (19)
C^∞ function: A c^∞ function, or a smooth function, is a function that is infinitely differentiable, meaning it has derivatives of all orders that are continuous. This property ensures that the function behaves well in terms of calculus, allowing for operations like integration and differentiation without encountering issues like discontinuities or undefined behavior. In the context of smooth maps, c^∞ functions are foundational as they maintain the structure needed for many important properties in differential topology.
C^k function: A c^k function is a function that is k-times continuously differentiable, meaning it has continuous derivatives up to order k. This property is significant in the study of smooth maps, as it allows for the analysis of various mathematical structures and behaviors that depend on the smoothness of the functions involved. The smoothness indicated by c^k differentiability ensures that not only the function itself is well-behaved, but also its derivatives, which can be crucial when considering concepts like continuity, limits, and mappings between different spaces.
Composition of smooth functions: The composition of smooth functions refers to the process of combining two smooth functions to create a new function. If you have two smooth functions, say $f: U \subseteq \mathbb{R}^n \to \mathbb{R}^m$ and $g: V \subseteq \mathbb{R}^m \to \mathbb{R}^p$, their composition $g \circ f$ is defined on the set $U$ as $(g \circ f)(x) = g(f(x))$ for all $x \in U$. This operation preserves the smoothness of the functions, meaning that if both $f$ and $g$ are smooth, then the resulting function $g \circ f$ is also smooth, which is a fundamental aspect when discussing smooth maps and their properties.
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.
Diffeomorphism: A diffeomorphism is a smooth, invertible map between two manifolds that has a smooth inverse. This concept is crucial for understanding when two manifolds can be considered 'the same' in terms of their smooth structure, as it allows for a rigorous notion of equivalence between them. Diffeomorphisms preserve the differential structure and are used to relate different types of manifolds like spheres, tori, and projective spaces to one another while retaining their topological features.
Diffeomorphism group: The diffeomorphism group is the collection of all diffeomorphisms from a smooth manifold to itself, where a diffeomorphism is a smooth, invertible function with a smooth inverse. This group captures the idea of smooth transformations that preserve the manifold's structure, allowing for the exploration of geometric and topological properties. The diffeomorphism group is crucial in understanding how smooth maps behave and interact, highlighting important concepts such as equivalence of shapes and structures in differential topology.
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.
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.
Index of a critical point: The index of a critical point is an integer that characterizes the local behavior of a smooth map near that point, specifically indicating the number of directions in which the map decreases versus those in which it increases. This index is essential for understanding the topology of manifolds and plays a crucial role in classifying critical points, especially in the context of Morse functions. It connects local analysis to global topological properties, offering insight into the nature of critical points.
Inverse Function Theorem: The Inverse Function Theorem states that if a function is continuously differentiable and its derivative is non-zero at a point, then it has a continuous inverse function near that point. This theorem plays a crucial role in understanding the behavior of smooth maps and their properties, as it provides conditions under which we can locally reverse mappings between spaces.
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.
Morse theory: Morse theory is a branch of mathematics that studies the topology of manifolds using smooth functions, particularly focusing on the critical points of these functions and their implications for the manifold's structure. By analyzing how these critical points behave under variations of the function, Morse theory connects the geometry of the manifold with its topology, providing deep insights into the shape and features of the space.
Regular Value: A regular value is a point in the target space of a smooth map such that the preimage of that point consists only of points where the differential of the map is surjective. This concept is important for understanding how smooth maps behave and has applications in various areas, including the implicit function theorem, submersions, and determining properties like the degree of a map.
Sard's Theorem: Sard's Theorem states that the set of critical values of a smooth map between manifolds has measure zero. This means that, when mapping from one manifold to another, most points in the target manifold are regular values, which helps understand how the smooth map behaves. The implications of this theorem stretch into various areas, such as the properties of submersions and regular values, influencing the topology and geometry involved in differentiable maps.
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.
Submersion: Submersion is a type of smooth map between differentiable manifolds where the differential of the map is surjective at every point in its domain. This means that for each point in the target manifold, there are points in the source manifold that are mapped to it, allowing for a rich structure in differential topology and the exploration of properties like regular values and smoothness.
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.
Whitney Immersion Theorem: The Whitney Immersion Theorem states that any smooth manifold can be immersed in Euclidean space of sufficiently high dimension. This result is significant because it establishes the conditions under which manifolds can be represented within a higher-dimensional space, which is crucial for understanding their geometric and topological properties. Additionally, the theorem has important implications for the study of smooth maps, as it provides a foundation for analyzing how these maps can behave when transferring between different dimensional spaces.