Coordinate charts and atlases are essential tools in differential geometry for describing manifolds. They provide a way to represent complex geometric structures using familiar coordinate systems, allowing for local analysis and calculations.
These concepts form the foundation for studying smooth manifolds. By using coordinate charts and atlases, we can define smoothness, perform computations, and analyze the geometric properties of manifolds in a systematic way.
Coordinate charts
Coordinate charts are a fundamental concept in differential geometry that allow us to describe a manifold locally using a set of coordinates
They provide a way to parametrize a portion of a manifold using a between an open subset of the manifold and an open subset of Euclidean space
Coordinate charts are essential for performing calculations and analyzing the geometric properties of manifolds
Definition of coordinate chart
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A on a topological manifold M is a pair (U,φ), where U is an open subset of M and φ:U→φ(U)⊂Rn is a homeomorphism
The map φ assigns coordinates to each point in U, providing a local coordinate system
The U is called the chart domain, and the homeomorphism φ is called the chart map or coordinate map
Homeomorphisms in coordinate charts
The chart map φ in a coordinate chart is required to be a homeomorphism, which means it is a continuous bijection with a continuous inverse
Homeomorphisms preserve the topological properties of the manifold, such as continuity and connectedness
The existence of a homeomorphism between the chart domain and an open subset of Euclidean space ensures that the local structure of the manifold is preserved under the coordinate representation
Transition maps between charts
When two coordinate charts (U,φ) and (V,ψ) overlap, i.e., U∩V=∅, we can define a transition map between them
The transition map is given by the composition ψ∘φ−1:φ(U∩V)→ψ(U∩V), which relates the coordinates of points in the overlap region
Transition maps are essential for ensuring consistency and compatibility between different coordinate charts on a manifold
Atlases
An atlas is a collection of coordinate charts that cover the entire manifold
Atlases provide a way to describe the global structure of a manifold by patching together local coordinate descriptions
The concept of an atlas is central to the study of differentiable manifolds and is used to define smooth structures on manifolds
Definition of atlas
An atlas on a topological manifold M is a collection of coordinate charts {(Uα,φα)}α∈A such that:
The chart domains Uα cover the entire manifold, i.e., ⋃α∈AUα=M
Any two charts in the atlas are compatible, meaning that the transition maps between overlapping charts are smooth (infinitely differentiable)
An atlas provides a complete coordinate description of the manifold, allowing for the study of its geometric and topological properties
Compatibility of charts in atlas
Compatibility of charts in an atlas ensures that the transition maps between overlapping charts are smooth
If two charts (U,φ) and (V,ψ) overlap, the transition map ψ∘φ−1:φ(U∩V)→ψ(U∩V) must be a smooth function
Compatibility allows for the consistent definition of smooth functions, vector fields, and other geometric objects on the manifold
Charts in an atlas are often referred to as "smoothly compatible" or "C∞-compatible" to emphasize the smoothness requirement
Maximal atlas
A maximal atlas on a manifold M is an atlas that contains all possible charts that are compatible with the charts already in the atlas
Given an atlas A on M, the maximal atlas containing A is the union of all charts that are compatible with every chart in A
The maximal atlas is unique and provides the most comprehensive coordinate description of the manifold
The existence of a maximal atlas is guaranteed by the axiom of choice and is used in the definition of smooth manifolds
Smooth structures
A smooth structure on a manifold is a maximal atlas that defines the notion of smoothness for functions and maps on the manifold
Smooth structures allow for the study of differential geometric properties, such as tangent spaces, vector fields, and differential forms
The choice of a smooth structure on a manifold is not unique, and different smooth structures can give rise to non-diffeomorphic smooth manifolds
Smooth atlas
A on a manifold M is an atlas in which all the transition maps between overlapping charts are smooth (infinitely differentiable)
A smooth atlas defines a smooth structure on the manifold, allowing for the consistent definition of smooth functions and maps
The existence of a smooth atlas is a prerequisite for studying the differential geometric properties of a manifold
Smooth manifolds
A smooth manifold is a pair (M,A), where M is a topological manifold and A is a smooth atlas on M
Smooth manifolds are the primary objects of study in differential geometry and provide a framework for analyzing geometric properties using calculus and linear algebra
Examples of smooth manifolds include Euclidean spaces, spheres, tori, and Lie groups
Smooth functions and maps
A function f:M→R on a smooth manifold M is called smooth if for every chart (U,φ) in the smooth atlas, the composition f∘φ−1:φ(U)→R is a smooth function on the open subset φ(U)⊂Rn
A map F:M→N between two smooth manifolds M and N is called smooth if for every pair of charts (U,φ) on M and (V,ψ) on N such that F(U)⊂V, the composition ψ∘F∘φ−1:φ(U)→ψ(V) is a smooth function
Smooth functions and maps are the building blocks for studying the differential geometric properties of manifolds, such as tangent spaces, vector fields, and differential forms
Coordinate representations
Coordinate representations allow us to express geometric objects and properties of a manifold in terms of
By using coordinate charts, we can represent points, curves, functions, and other geometric objects as tuples of real numbers or functions of real variables
Coordinate representations simplify calculations and provide a way to apply analytical methods to the study of manifolds
Coordinate representation of points
Given a coordinate chart (U,φ) on a manifold M and a point p∈U, the coordinate representation of p is the tuple φ(p)=(x1(p),…,xn(p))∈Rn
The components xi(p) are called the coordinates of p with respect to the chart (U,φ)
Coordinate representations of points allow us to identify points on the manifold with tuples of real numbers, enabling the use of analytical methods
Coordinate representation of curves
A curve on a manifold M is a smooth map γ:I→M, where I⊂R is an open interval
Given a coordinate chart (U,φ) on M, the coordinate representation of a curve γ in U is the composition φ∘γ:I→φ(U)⊂Rn
The coordinate representation of a curve is a vector-valued function of a real variable, allowing for the study of its properties using calculus
Coordinate representation of functions
A function f:M→R on a manifold M can be expressed in local coordinates using a coordinate chart (U,φ)
The coordinate representation of f in U is the composition f∘φ−1:φ(U)→R, which is a function of n real variables, where n is the of the manifold
Coordinate representations of functions allow for the application of analytical methods, such as differentiation and integration, to the study of functions on manifolds
Computations in coordinates
Computations in coordinates involve expressing geometric objects and operations in terms of local coordinate representations
By using coordinate charts, we can perform calculations involving vector fields, differential forms, and other geometric quantities using the tools of linear algebra and calculus
Computations in coordinates are essential for solving problems and deriving properties of manifolds and their associated geometric structures
Vector fields in coordinates
A vector field on a manifold M is a smooth assignment of a tangent vector to each point of the manifold
In local coordinates, a vector field X can be expressed as a linear combination of the coordinate basis vectors: X=Xi∂xi∂
The components Xi are functions of the local coordinates and determine the behavior of the vector field in the given coordinate chart
Coordinate representations of vector fields allow for the study of their properties, such as divergence, curl, and Lie derivatives
Differential forms in coordinates
A differential form on a manifold M is a smooth assignment of an alternating multilinear map on the tangent spaces of M
In local coordinates, a differential k-form ω can be expressed as a linear combination of the coordinate basis forms: ω=ωi1…ikdxi1∧…∧dxik
The components ωi1…ik are functions of the local coordinates and determine the behavior of the differential form in the given coordinate chart
Coordinate representations of differential forms allow for the study of their properties, such as exterior derivatives, wedge products, and integration
Christoffel symbols in coordinates
Christoffel symbols are the components of the Levi-Civita connection, which is a fundamental object in Riemannian geometry
In local coordinates, the Christoffel symbols Γijk are defined in terms of the metric tensor gij and its partial derivatives: Γijk=21gkl(∂xi∂gjl+∂xj∂gil−∂xl∂gij)
Christoffel symbols play a crucial role in the computation of geodesics, parallel transport, and curvature tensors in Riemannian manifolds
Coordinate representations of Christoffel symbols allow for the study of the geometric properties of Riemannian manifolds using the tools of tensor calculus
Coordinate transformations
Coordinate transformations describe how geometric objects and properties change when transitioning between different coordinate charts on a manifold
Understanding coordinate transformations is essential for ensuring the consistency and invariance of geometric concepts across different coordinate representations
Coordinate transformations involve the Jacobian matrix, change of variables formulas, and the transformation laws for tensors and other geometric objects
Jacobian matrix
The Jacobian matrix of a coordinate transformation φ:U→V between two coordinate charts (U,xi) and (V,yj) is the matrix of partial derivatives: Jijφ=∂xj∂yi
The Jacobian matrix encodes the local behavior of the coordinate transformation and is used to transform vectors, differential forms, and other geometric objects between coordinate charts
The determinant of the Jacobian matrix, called the Jacobian determinant, measures the local change in volume under the coordinate transformation
Change of variables
Change of variables formulas describe how integrals transform under a coordinate transformation
Given a coordinate transformation φ:U→V and a function f:V→R, the change of variables formula for integrals states: ∫Vf(y)dy=∫Uf(φ(x))∣detJφ(x)∣dx
Change of variables formulas are essential for computing integrals over manifolds and for deriving conservation laws and other integral identities
Transformation of metric tensor
The metric tensor g on a is a symmetric, positive-definite tensor field that defines the inner product on tangent spaces
Under a coordinate transformation φ:U→V, the components of the metric tensor transform according to the rule: gij′(y)=∂yi∂xk∂yj∂xlgkl(x)
The transformation law for the metric tensor ensures that the length of curves, angles between vectors, and other geometric quantities are preserved under coordinate transformations
Understanding the transformation properties of the metric tensor is crucial for studying the intrinsic geometry of Riemannian manifolds
Atlas constructions
Constructing atlases on manifolds is an important task in differential geometry, as it provides a way to define smooth structures and study the geometric properties of the manifold
There are several standard techniques for constructing atlases, including stereographic projection, the exponential map, and coordinate patches on spheres
These construction methods are often used to define smooth structures on common manifolds, such as spheres, projective spaces, and Lie groups
Stereographic projection
Stereographic projection is a method for constructing a coordinate chart on the sphere Sn by projecting points from the sphere onto a tangent plane
The stereographic projection map φ:Sn∖{N}→Rn is defined by projecting a point P on the sphere from the north pole N onto the equatorial plane
Stereographic projection provides a smooth coordinate chart on the sphere that covers all but one point (the north pole)
The stereographic projection atlas consists of two charts: the stereographic projection from the north pole and the stereographic projection from the south pole, which together cover the entire sphere
Exponential map
The exponential map is a method for constructing coordinate charts on a Riemannian manifold using geodesics
Given a point p on a Riemannian manifold M, the exponential map expp:TpM→M maps a tangent vector v at p to the point on the manifold obtained by following the geodesic starting at p with initial velocity v for a unit time
The exponential map provides a local between a neighborhood of the origin in the tangent space TpM and a neighborhood of p on the manifold
Exponential maps are used to construct local coordinate charts, called normal coordinates, which are particularly useful for studying the geometry of the manifold near a given point
Coordinate patches on spheres
Coordinate patches on spheres are a way to construct atlases on spheres of arbitrary dimension using a combination of stereographic projections and rotations
The standard coordinate patches on the unit sphere Sn are defined using stereographic projections from the north and south poles, along with rotations that move the poles to other points on the sphere
For example, on the 2-sphere S2, the standard coordinate patches consist of six charts: the stereographic projections from the north and south poles, and the stereographic projections from the four points obtained by rotating the poles by 90 degrees around the coordinate axes
These coordinate patches provide a smooth atlas on the sphere and are often used to study the geometry and topology of spherical manifolds
Key Terms to Review (17)
Change of Coordinates: Change of coordinates refers to the process of expressing a mathematical object, such as a point or vector, in terms of different coordinate systems. This process is essential in understanding how various geometric structures can be analyzed from multiple perspectives, facilitating the study of properties that remain invariant under such transformations.
Coordinate Chart: A coordinate chart is a mathematical tool used in differential geometry that assigns coordinates to points in a manifold, facilitating the description of its geometric properties. It provides a way to translate abstract concepts of manifolds into more familiar settings, enabling calculations and analysis through local coordinates. The concept is crucial for understanding how manifolds can be covered by collections of charts, known as atlases, that together give a complete picture of the manifold's structure.
Diffeomorphism: A diffeomorphism is a smooth, invertible function between two differentiable manifolds that has a smooth inverse. This concept is crucial in understanding how manifolds relate to each other and allows for the comparison of their geometric structures. Diffeomorphisms preserve the manifold's differentiable structure, making them essential for analyzing properties like tangent spaces and induced metrics when considering submanifolds.
Differentiable Atlas: A differentiable atlas is a collection of coordinate charts that are smoothly compatible, meaning that the transition maps between the charts are differentiable. This concept is crucial for understanding how different parts of a manifold can be described using various coordinate systems while ensuring that calculations and geometric properties can be consistently analyzed across these systems. The smooth compatibility ensures that the manifold has a well-defined differential structure.
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.
Homeomorphism: A homeomorphism is a continuous function between two topological spaces that has a continuous inverse, essentially providing a way to show that two spaces are 'the same' in a topological sense. This concept plays a crucial role in understanding the properties of spaces, as it indicates that these spaces can be transformed into one another without tearing or gluing, preserving their topological characteristics. The significance of homeomorphisms can be seen in various mathematical contexts, from defining equivalence classes of shapes to analyzing structures like manifolds and submanifolds.
Local coordinates: Local coordinates are a system of numerical labels that describe points within a small neighborhood of a manifold, allowing for calculations and analysis in a simplified context. They enable the transition between different coordinate systems and provide a way to understand the geometric and topological properties of the manifold in a manageable manner. By using local coordinates, one can express curves, surfaces, and other structures more conveniently, making them essential for various mathematical frameworks.
N-dimensional manifold: An n-dimensional manifold is a topological space that locally resembles Euclidean space of dimension n, meaning that around every point, there is a neighborhood that can be mapped to an open subset of Euclidean space. This concept allows for the generalization of geometric and topological ideas from flat spaces to more complex shapes, enabling the study of objects that may not be globally Euclidean but have manageable local properties.
Open Set: An open set is a fundamental concept in topology, defined as a set where, for every point within it, there exists a neighborhood around that point that is entirely contained within the set. This concept is crucial as it lays the groundwork for defining continuous functions, convergence, and other essential properties in a topological space. In the context of coordinate charts and atlases, open sets help define the local structure of manifolds and facilitate smooth transitions between different coordinate systems.
Polar Coordinates: Polar coordinates provide a way to describe the position of points in a two-dimensional plane using a distance from a reference point and an angle from a reference direction. This system differs from Cartesian coordinates by focusing on the radial distance and angle, which is especially useful in situations involving circular or rotational symmetry, making connections to coordinate charts and atlases, as well as transition maps and compatibility.
Riemannian manifold: A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which allows for the measurement of lengths of curves, angles between vectors, and areas of surfaces. This structure provides the geometric framework needed to study concepts like curvature and distance in a way that generalizes the familiar properties of Euclidean space.
Smooth atlas: A smooth atlas is a collection of smooth coordinate charts that cover a manifold, allowing for the transition between charts to be smooth functions. This concept is crucial as it ensures that the manifold can be analyzed and understood in a consistent way using different coordinate systems. The smooth atlas is fundamental in the study of differential geometry, providing the structure needed to define smoothness on manifolds and facilitating the study of their geometric properties.
Spherical coordinates: Spherical coordinates is a three-dimensional coordinate system that represents points in space using three parameters: the radial distance from a fixed origin, the polar angle (or colatitude) from the positive z-axis, and the azimuthal angle (or longitude) in the x-y plane. This system is particularly useful in scenarios where problems exhibit spherical symmetry, as it simplifies the equations and calculations involved.
Submersion and Immersion: Submersion and immersion refer to types of smooth mappings between manifolds, focusing on the relationship between their dimensions. Specifically, a submersion is a smooth map where the differential is surjective, while an immersion is a smooth map where the differential is injective. These concepts are crucial for understanding how coordinate charts and atlases describe manifold structures and how they interact with each other through smooth transitions.
Symplectic manifold: A symplectic manifold is a smooth, even-dimensional manifold equipped with a closed, non-degenerate 2-form known as the symplectic form. This structure allows for the study of geometric properties related to Hamiltonian mechanics and provides a framework for understanding the conservation laws and dynamical systems associated with physical systems. The symplectic structure is crucial for defining Hamiltonian dynamics, where the phase space of a system can be modeled as a symplectic manifold.
Topological space: A topological space is a set equipped with a collection of open subsets that satisfy certain properties, allowing for the formalization of concepts like continuity, convergence, and compactness. This framework provides a way to analyze the structure and properties of spaces without requiring a specific geometric form. Understanding topological spaces is essential for discussing how different spaces can be mapped or transformed while preserving their essential characteristics.
Transition Function: A transition function is a mathematical mapping that describes how to change from one coordinate chart to another on a manifold. It plays a crucial role in ensuring that different charts provide consistent and compatible descriptions of the manifold's structure, allowing us to move between various representations seamlessly. Transition functions are essential for defining smoothness and differentiability in the context of differential geometry.