1.3 Coordinate charts and atlases

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 homeomorphism 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

• A coordinate chart on a topological manifold $M$ is a pair $(U, \varphi)$, where $U$ is an open subset of $M$ and $\varphi: U \to \varphi(U) \subset \mathbb{R}^n$ is a homeomorphism
• The map $\varphi$ assigns coordinates to each point in $U$, providing a local coordinate system
• The open set $U$ is called the chart domain, and the homeomorphism $\varphi$ is called the chart map or coordinate map

Homeomorphisms in coordinate charts

• The chart map $\varphi$ 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 $(U, \varphi)$ and $(V, \psi)$ overlap, i.e., $U \cap V \neq \emptyset$, we can define a transition map between them
• The transition map is given by the composition $\psi \circ \varphi^{-1}: \varphi(U \cap V) \to \psi(U \cap V)$, 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 $\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}$ such that:
  1. The chart domains $U_\alpha$ cover the entire manifold, i.e., $\bigcup_{\alpha \in A} U_\alpha = M$
  2. 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 $(U, \varphi)$ and $(V, \psi)$ overlap, the transition map $\psi \circ \varphi^{-1}: \varphi(U \cap V) \to \psi(U \cap V)$ 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 "$C^\infty$-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 $\mathcal{A}$ on $M$, the maximal atlas containing $\mathcal{A}$ is the union of all charts that are compatible with every chart in $\mathcal{A}$
• 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 smooth atlas 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 $(M, \mathcal{A})$, where $M$ is a topological manifold and $\mathcal{A}$ 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 $f: M \to \mathbb{R}$ on a smooth manifold $M$ is called smooth if for every chart $(U, \varphi)$ in the smooth atlas, the composition $f \circ \varphi^{-1}: \varphi(U) \to \mathbb{R}$ is a smooth function on the open subset $\varphi(U) \subset \mathbb{R}^n$
• A map $F: M \to N$ between two smooth manifolds $M$ and $N$ is called smooth if for every pair of charts $(U, \varphi)$ on $M$ and $(V, \psi)$ on $N$ such that $F(U) \subset V$, the composition $\psi \circ F \circ \varphi^{-1}: \varphi(U) \to \psi(V)$ 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 local coordinates
• 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 $(U, \varphi)$ on a manifold $M$ and a point $p \in U$, the coordinate representation of $p$ is the tuple $\varphi(p) = (x^1(p), \ldots, x^n(p)) \in \mathbb{R}^n$
• The components $x^i(p)$ are called the coordinates of $p$ with respect to the chart $(U, \varphi)$
• 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 $\gamma: I \to M$, where $I \subset \mathbb{R}$ is an open interval
• Given a coordinate chart $(U, \varphi)$ on $M$, the coordinate representation of a curve $\gamma$ in $U$ is the composition $\varphi \circ \gamma: I \to \varphi(U) \subset \mathbb{R}^n$
• 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 $f: M \to \mathbb{R}$ on a manifold $M$ can be expressed in local coordinates using a coordinate chart $(U, \varphi)$
• The coordinate representation of $f$ in $U$ is the composition $f \circ \varphi^{-1}: \varphi(U) \to \mathbb{R}$, which is a function of $n$ real variables, where $n$ is the dimension 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: $X = X^i \frac{\partial}{\partial x^i}$
• The components $X^i$ 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 $\omega$ can be expressed as a linear combination of the coordinate basis forms: $\omega = \omega_{i_1 \ldots i_k} dx^{i_1} \wedge \ldots \wedge dx^{i_k}$
• The components $\omega_{i_1 \ldots i_k}$ 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 $\Gamma^k_{ij}$ are defined in terms of the metric tensor $g_{ij}$ and its partial derivatives: $\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left(\frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{il}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^l}\right)$
• 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 $\varphi: U \to V$ between two coordinate charts $(U, x^i)$ and $(V, y^j)$ is the matrix of partial derivatives: $J^\varphi_{ij} = \frac{\partial y^i}{\partial x^j}$
• 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 $\varphi: U \to V$ and a function $f: V \to \mathbb{R}$, the change of variables formula for integrals states: $\int_V f(y) dy = \int_U f(\varphi(x)) |\det J^\varphi(x)| dx$
• 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 Riemannian manifold is a symmetric, positive-definite tensor field that defines the inner product on tangent spaces
• Under a coordinate transformation $\varphi: U \to V$, the components of the metric tensor transform according to the rule: $g'_{ij}(y) = \frac{\partial x^k}{\partial y^i} \frac{\partial x^l}{\partial y^j} g_{kl}(x)$
• 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 $S^n$ by projecting points from the sphere onto a tangent plane
• The stereographic projection map $\varphi: S^n \setminus \{N\} \to \mathbb{R}^n$ 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 $\exp_p: T_pM \to M$ 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 diffeomorphism between a neighborhood of the origin in the tangent space $T_pM$ 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 $S^n$ 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 $S^2$, 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