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 smooth structure, 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

• 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 chart on a manifold is a pair $(U, \varphi)$, where $U$ is an open subset of the manifold and $\varphi: U \to V$ is a homeomorphism onto an open subset $V$ of Euclidean space
• An atlas 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 $(U, \varphi)$ and $(V, \psi)$, the transition map between them is the composition $\psi \circ \varphi^{-1}: \varphi(U \cap V) \to \psi(U \cap V)$
• 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 $\mathbb{R}^n$ are the simplest examples of smooth manifolds
• The standard topology and smooth structure on $\mathbb{R}^n$ are given by the identity chart, making it a smooth manifold of dimension $n$
• Euclidean spaces serve as local models for general smooth manifolds through the use of charts

Spheres and tori

• The $n$-sphere $S^n$ is the set of points in $\mathbb{R}^{n+1}$ at a fixed distance from the origin, inheriting a smooth structure as a submanifold of Euclidean space
• Tori, such as the 2-torus $T^2$, 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 $GL(n, \mathbb{R})$ 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 $\mathbb{R}^n$, carrying a natural smooth structure
• Grassmann manifolds are important in algebraic geometry and optimization, representing spaces of linear constraints or subspaces
• The projective space $\mathbb{RP}^n$ is a special case of a Grassmann manifold, representing lines through the origin in $\mathbb{R}^{n+1}$

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 $f: M \to N$ between smooth manifolds is smooth if for every point $p \in M$, there exist charts $(U, \varphi)$ around $p$ and $(V, \psi)$ around $f(p)$ such that the composition $\psi \circ f \circ \varphi^{-1}$ is a smooth function 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 smooth map 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 tangent space 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 $T_pM$ 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 $T_p^M$ 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 symplectic structure, 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 $d^2 = 0$ 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 $i: S \to M$ 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 Riemannian metric, 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., $GL(n, \mathbb{R}), O(n), U(n)$), as well as more abstract groups such as the Heisenberg group and the Poincaré group