Smooth manifolds are the backbone of differential geometry, extending the concept of curves and surfaces to higher dimensions. They provide a framework for applying calculus and analysis to complex geometric spaces, enabling the study of intricate mathematical structures.
In this chapter, we explore the definition and properties of smooth manifolds, including their topological foundations and smooth structures. We'll examine various examples, from simple Euclidean spaces to more complex objects like spheres and matrix Lie groups, to build intuition about these fundamental mathematical entities.
Definition of smooth manifolds
Smooth manifolds are central objects of study in differential geometry, generalizing curves and surfaces to higher dimensions
A smooth manifold is a topological space equipped with a , allowing for calculus and analysis to be performed on the manifold
Smooth manifolds provide a framework for studying geometric and topological properties of spaces using tools from calculus and linear algebra
Topological spaces as manifolds
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A topological space is a set equipped with a collection of open sets satisfying certain axioms
For a topological space to be a manifold, it must be locally Euclidean, meaning each point has a neighborhood homeomorphic to an open subset of Euclidean space
Manifolds are required to be Hausdorff and second-countable, ensuring they have well-behaved topological properties
Smooth structures on manifolds
A smooth structure on a manifold is a collection of charts that are smoothly compatible with each other
Charts are homeomorphisms from open subsets of the manifold to open subsets of Euclidean space
The smooth structure allows for the definition of smooth functions and the use of calculus on the manifold
Charts and atlases
A on a manifold is a pair (U,φ), where U is an open subset of the manifold and φ:U→V is a homeomorphism onto an open subset V of Euclidean space
An is a collection of charts that cover the entire manifold
The charts in an atlas must be smoothly compatible, meaning the transition maps between overlapping charts are smooth functions
Transition maps between charts
Given two overlapping charts (U,φ) and (V,ψ), the transition map between them is the composition ψ∘φ−1:φ(U∩V)→ψ(U∩V)
For a smooth manifold, the transition maps between charts must be smooth functions
The smoothness of transition maps ensures that the notion of smoothness is well-defined and independent of the choice of charts
Examples of smooth manifolds
Smooth manifolds are abundant in mathematics and physics, with many important spaces naturally carrying a smooth structure
Studying examples of smooth manifolds helps develop intuition and understanding of their properties and behavior
Euclidean spaces as manifolds
Euclidean spaces Rn are the simplest examples of smooth manifolds
The standard topology and smooth structure on Rn are given by the identity chart, making it a smooth manifold of n
Euclidean spaces serve as local models for general smooth manifolds through the use of charts
Spheres and tori
The n-sphere Sn is the set of points in Rn+1 at a fixed distance from the origin, inheriting a smooth structure as a submanifold of Euclidean space
Tori, such as the 2-torus T2, are obtained as products of circles, carrying a natural smooth structure
Spheres and tori are important examples of compact smooth manifolds with non-trivial topology
Matrix Lie groups
Matrix Lie groups, such as the general linear group GL(n,R) and the special orthogonal group SO(n), are smooth manifolds
The smooth structure on matrix Lie groups is induced by the smooth structure on the space of matrices
Matrix Lie groups play a central role in the study of symmetries and transformations in mathematics and physics
Grassmann manifolds
The Grassmann manifold Gr(k,n) is the set of k-dimensional linear subspaces of Rn, carrying a natural smooth structure
Grassmann manifolds are important in algebraic geometry and optimization, representing spaces of linear constraints or subspaces
The projective space RPn is a special case of a Grassmann manifold, representing lines through the origin in Rn+1
Smooth maps between manifolds
Smooth maps between manifolds are the morphisms in the category of smooth manifolds, preserving the smooth structure
Studying smooth maps allows for the comparison and relation of different manifolds, as well as the transfer of geometric and topological properties
Definition of smooth maps
A map f:M→N between smooth manifolds is smooth if for every point p∈M, there exist charts (U,φ) around p and (V,ψ) around f(p) such that the composition ψ∘f∘φ−1 is a between open subsets of Euclidean spaces
Smoothness of maps is a local property, depending only on the behavior in small neighborhoods of points
Smooth maps are continuous and infinitely differentiable, allowing for the use of calculus techniques
Diffeomorphisms and embeddings
A diffeomorphism is a with a smooth inverse, providing an isomorphism between smooth manifolds
Diffeomorphic manifolds have the same smooth structure and can be considered equivalent from the perspective of differential geometry
An embedding is a smooth map that is a diffeomorphism onto its image, realizing one manifold as a submanifold of another
Immersions and submersions
An immersion is a smooth map with everywhere injective differential, locally embedding the domain manifold into the codomain
A submersion is a smooth map with everywhere surjective differential, locally projecting the domain manifold onto the codomain
Immersions and submersions are important in the study of foliations and fiber bundles on manifolds
Inverse function theorem
The inverse function theorem states that a smooth map with everywhere invertible differential is locally a diffeomorphism
The theorem provides a powerful tool for constructing local inverses and charts on manifolds
Applications of the inverse function theorem include the implicit function theorem and the constant rank theorem
Tangent spaces and vectors
Tangent spaces and vectors are the fundamental linear approximations to a manifold at each point, encoding infinitesimal information about the smooth structure
The at a point can be thought of as the space of velocity vectors of curves passing through that point
Tangent spaces at points
The tangent space TpM at a point p on a manifold M is a vector space of the same dimension as M
Tangent spaces can be defined equivalently as equivalence classes of curves through p or as derivations on the algebra of smooth functions at p
The disjoint union of all tangent spaces forms the tangent bundle TM of the manifold
Tangent vectors as derivations
A tangent vector at a point p can be defined as a derivation on the algebra of germs of smooth functions at p
Derivations are linear maps satisfying the Leibniz rule, capturing the idea of directional derivatives
The derivation perspective provides a coordinate-free approach to tangent vectors and is useful in algebraic geometry
Tangent bundles of manifolds
The tangent bundle TM of a manifold M is the disjoint union of all tangent spaces, carrying a natural smooth structure
The tangent bundle is a vector bundle over M, with each fiber being the tangent space at the corresponding point
Sections of the tangent bundle are vector fields on the manifold, assigning a tangent vector to each point
Vector fields on manifolds
A vector field on a manifold M is a smooth assignment of a tangent vector to each point of M
Vector fields can be viewed as derivations on the algebra of smooth functions on M, generalizing the notion of directional derivatives
The space of vector fields on a manifold forms a Lie algebra under the Lie bracket operation, capturing the non-commutativity of flows
Cotangent spaces and forms
Cotangent spaces and forms are the dual notions to tangent spaces and vectors, providing a way to measure and integrate on manifolds
Differential forms are antisymmetric multilinear functionals on tangent spaces, generalizing the concept of integration
Dual spaces and covectors
The dual space Tp∗M of the tangent space at a point p is called the cotangent space, consisting of linear functionals on tangent vectors
Elements of the cotangent space are called covectors or 1-forms, generalizing the notion of differentials of functions
The disjoint union of all cotangent spaces forms the cotangent bundle T∗M of the manifold
Cotangent bundles of manifolds
The cotangent bundle T∗M of a manifold M is the vector bundle whose fiber at each point is the cotangent space at that point
Sections of the cotangent bundle are differential 1-forms on the manifold, assigning a covector to each point
The cotangent bundle carries a natural , making it a fundamental object in symplectic geometry
Differential forms on manifolds
A differential k-form on a manifold M is a smooth assignment of an alternating multilinear functional on the tangent space at each point
Differential forms provide a way to integrate over submanifolds of M, generalizing the notion of integration of functions
The exterior derivative d maps k-forms to (k+1)-forms, satisfying d2=0 and leading to the de Rham cohomology of the manifold
Exterior algebra of forms
The exterior algebra of forms on a manifold M is the graded algebra generated by differential forms under the wedge product
The wedge product is an antisymmetric multiplication operation, satisfying graded commutativity and associativity
The exterior algebra provides a natural setting for the study of differential forms and their properties, such as the Hodge star operator and the Lie derivative
Submanifolds and embeddings
Submanifolds are manifolds that are embedded or immersed inside other manifolds, inheriting a smooth structure from the ambient space
The study of submanifolds is important in understanding the local and global geometry of manifolds, as well as in applications such as constrained optimization
Definition of submanifolds
A subset S of a manifold M is a submanifold if it is a manifold in its own right and the inclusion map i:S→M is an immersion
The dimension of a submanifold is always less than or equal to the dimension of the ambient manifold
Examples of submanifolds include open subsets of manifolds, level sets of submersions, and graphs of smooth functions
Embedded vs immersed submanifolds
An embedded submanifold is a submanifold for which the inclusion map is a homeomorphism onto its image
An immersed submanifold is a submanifold for which the inclusion map is only locally a homeomorphism onto its image
Embedded submanifolds are more rigid and have nicer topological properties, while immersed submanifolds allow for self-intersections and more flexible geometry
Normal bundles of submanifolds
The normal bundle of a submanifold S in a Riemannian manifold M is the orthogonal complement of the tangent bundle of S in the restricted tangent bundle of M
The normal bundle encodes the extrinsic geometry of the submanifold, measuring how it sits inside the ambient manifold
Sections of the normal bundle are normal vector fields, which are important in the study of the second fundamental form and curvature of submanifolds
Tubular neighborhood theorem
The tubular neighborhood theorem states that every embedded submanifold of a manifold has a neighborhood that is diffeomorphic to a neighborhood of the zero section in its normal bundle
The theorem provides a local model for the geometry of the submanifold and its relationship to the ambient manifold
Tubular neighborhoods are useful in the study of the topology of submanifolds, as well as in the construction of collars and isotopies
Smooth manifolds with additional structure
Smooth manifolds can be equipped with additional geometric structures, leading to rich theories and applications
Examples of such structures include Riemannian metrics, symplectic forms, complex structures, and Lie group actions
Riemannian manifolds and metrics
A Riemannian manifold is a smooth manifold equipped with a , which is a smooth assignment of an inner product to each tangent space
Riemannian metrics allow for the measurement of lengths, angles, and volumes on the manifold, as well as the definition of geodesics and curvature
The study of Riemannian manifolds is central to differential geometry and has applications in physics, such as general relativity and mechanics
Symplectic manifolds and forms
A symplectic manifold is a smooth manifold equipped with a symplectic form, which is a closed, non-degenerate 2-form
Symplectic forms provide a natural setting for Hamiltonian mechanics and the study of conservative dynamical systems
Symplectic geometry is a rich and active area of research, with connections to algebraic geometry, topology, and mathematical physics
Complex manifolds and charts
A complex manifold is a smooth manifold equipped with an atlas of charts taking values in open subsets of complex Euclidean space, with holomorphic transition maps
Complex manifolds are the natural setting for the study of holomorphic functions and complex geometry
Examples of complex manifolds include Riemann surfaces, complex projective spaces, and complex Lie groups
Lie groups as smooth manifolds
A Lie group is a smooth manifold that is also a group, with smooth group operations of multiplication and inversion
Lie groups provide a natural framework for studying continuous symmetries and transformations in mathematics and physics
Examples of Lie groups include matrix groups (e.g., GL(n,R),O(n),U(n)), as well as more abstract groups such as the Heisenberg group and the Poincaré group
Key Terms to Review (17)
Atlas: In differential geometry, an atlas is a collection of charts that together cover a manifold. Each chart consists of a homeomorphism from an open subset of the manifold to an open subset of a Euclidean space, allowing for smooth transitions between different parts of the manifold. This concept is essential for defining smooth structures on manifolds and facilitates the understanding of their geometric and topological properties.
Chart: In the context of smooth manifolds, a chart is a mathematical tool that provides a way to describe a manifold by mapping a portion of it to an open subset of Euclidean space. This mapping is called a coordinate chart and helps in understanding the manifold's structure by allowing the use of coordinates to facilitate calculations and analysis. Charts are essential for defining smooth functions, differentiability, and the overall topology of the manifold.
Codimension: Codimension is defined as the difference between the dimension of a manifold and the dimension of a submanifold within it. This concept helps to understand how 'large' or 'small' a submanifold is relative to the manifold it resides in. It plays a critical role in various fields such as topology and differential geometry, highlighting the structure and properties of smooth manifolds and how they can be partitioned into lower-dimensional spaces.
Coordinate Patch: A coordinate patch is a mapping from an open subset of a manifold to an open subset of Euclidean space, allowing us to assign coordinates to points on the manifold. This mapping is a crucial tool for studying the properties of smooth manifolds and parametrized surfaces, as it enables us to express geometric and topological features in terms of familiar coordinate systems. Coordinate patches help create a bridge between abstract mathematical objects and concrete numerical representations.
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.
Differentiable Map: A differentiable map is a function between two smooth manifolds that preserves the smooth structure, meaning it can be expressed in local coordinates as a differentiable function. This concept is fundamental in understanding how manifolds interact and allows for the study of their geometric and topological properties. It plays a crucial role in the analysis of smooth structures and is pivotal when discussing tangent spaces and differential forms.
Dimension: Dimension refers to the minimum number of coordinates needed to specify a point within a mathematical space. In geometry, it is a fundamental aspect that helps describe the structure and behavior of various spaces, such as smooth manifolds, coordinate systems, and homogeneous spaces. The concept of dimension provides insight into how these spaces can be understood and navigated, establishing a framework for both local and global properties.
Pullback: In differential geometry, a pullback refers to the operation that takes a differential form defined on a manifold and allows it to be transported back to another manifold via a smooth map. This concept is crucial for understanding how properties of one manifold can relate to another, particularly when dealing with smooth manifolds, induced metrics on submanifolds, and conformal metrics. The pullback essentially enables us to analyze forms and functions in the context of their original manifolds, facilitating the study of geometric structures.
Pushforward: The pushforward is a concept in differential geometry that describes how a function between manifolds transforms tangent vectors at a point from the domain manifold to the target manifold. This transformation captures how the structure of the original manifold is carried over to another manifold under smooth mappings. The pushforward is essential for understanding how vector fields behave when they are moved from one space to another, as well as for analyzing geometric properties of metrics in different contexts.
Riemannian metric: A Riemannian metric is a smoothly varying positive definite inner product on the tangent spaces of a differentiable manifold, allowing one to measure distances and angles on that manifold. This concept forms the backbone of Riemannian geometry, enabling the exploration of curves, surfaces, and their properties in various contexts, from smooth manifolds to curvature concepts and beyond.
Smooth function: A smooth function is a mathematical function that is infinitely differentiable, meaning it has derivatives of all orders at every point in its domain. This property ensures that the function behaves nicely without any abrupt changes, allowing for the application of calculus techniques. Smooth functions are foundational in many areas of mathematics, particularly in the study of smooth manifolds and variational calculus, where their properties facilitate the analysis of geometric structures and physical systems.
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.
Smooth structure: A smooth structure on a manifold is a way to define differentiability on the manifold, allowing for the application of calculus to study its geometric properties. This structure consists of a collection of charts that are smoothly compatible with each other, enabling the transition between different local coordinate systems. The smooth structure is essential for understanding the manifold as a whole and connecting local properties to global characteristics.
Submersion Theorem: The Submersion Theorem states that if a smooth map between smooth manifolds is a submersion at a point, then the differential of the map is surjective at that point. This concept connects to the idea that a submersion locally resembles a projection, allowing for the exploration of lower-dimensional structures within higher-dimensional spaces. It highlights how smooth maps can create fibers and induce structures on manifolds, establishing a foundation for understanding differentiable mappings and their implications in differential geometry.
Symplectic Structure: A symplectic structure is a smooth, closed, non-degenerate 2-form defined on a smooth manifold, providing a geometric framework for the study of Hamiltonian mechanics. This structure allows one to analyze the behavior of dynamical systems and plays a crucial role in defining the phase space of these systems. It captures the essence of symplectic geometry by facilitating the formulation of physical laws governing conservation and transformations.
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.