🔁Elementary Differential Topology Unit 2 – Smooth Maps and Differentiable Functions

Key Concepts and Definitions

• Smooth maps are functions between smooth manifolds that preserve the differentiable structure and are infinitely differentiable
• Differentiable functions are functions between Euclidean spaces that have continuous partial derivatives up to a certain order
• Tangent spaces are vector spaces attached to each point of a manifold that capture the local linear approximation of the manifold
• Differential of a smooth map is a linear map between tangent spaces that captures the local behavior of the smooth map
• Diffeomorphisms are smooth maps with smooth inverses, establishing an equivalence between smooth manifolds
• Immersions are smooth maps with injective differential at every point, locally embedding the domain manifold into the codomain
• Submersions are smooth maps with surjective differential at every point, locally projecting the domain manifold onto the codomain

Smooth Maps: Fundamentals

• Smooth maps between smooth manifolds are defined using local coordinate charts and the notion of differentiability in Euclidean spaces
• Composition of smooth maps is smooth, allowing for the study of categories of smooth manifolds and smooth maps
• Smooth maps pull back smooth functions, providing a contravariant functor from the category of smooth manifolds to the category of algebras
  • Pullback of a smooth function $f$ by a smooth map $\varphi$ is given by $\varphi^*(f) = f \circ \varphi$
• Smooth maps push forward tangent vectors, providing a covariant functor from the category of smooth manifolds to the category of vector bundles
  • Pushforward of a tangent vector $v$ at a point $p$ by a smooth map $\varphi$ is given by $\varphi_*(v) = d\varphi_p(v)$
• Smooth maps preserve the dimension of manifolds, as a consequence of the rank theorem for differentiable functions
• Smooth maps can be locally approximated by their differential, which is a linear map between tangent spaces

Properties of Differentiable Functions

• Differentiable functions between Euclidean spaces have continuous partial derivatives up to a certain order
• Sum, product, and composition of differentiable functions are differentiable, with the order of differentiability determined by the minimum of the orders of the component functions
• Chain rule relates the differential of a composition to the differentials of the component functions: $d(g \circ f)_p = dg_{f(p)} \circ df_p$
• Inverse function theorem states that a differentiable function with non-singular differential at a point has a local differentiable inverse
  • Non-singular differential means that the Jacobian matrix is invertible
• Implicit function theorem allows for the local solution of systems of equations defined by differentiable functions
  • Provides conditions for the existence of implicit functions defined by level sets of differentiable functions
• Taylor's theorem approximates a differentiable function by a polynomial using higher-order derivatives
  • Remainder term in Taylor's theorem quantifies the error in the polynomial approximation

Examples and Counterexamples

• Identity map on any smooth manifold is a smooth map and a diffeomorphism
• Constant maps between smooth manifolds are smooth, but not necessarily immersions or submersions
• Inclusion map of a submanifold into a manifold is a smooth immersion
• Projection maps from product manifolds onto their factors are smooth submersions
• Stereographic projection is a diffeomorphism between the sphere minus a point and the Euclidean space of the same dimension
• Möbius strip is a smooth manifold that is not diffeomorphic to the cylinder, despite being locally similar
• Smooth functions with compact support provide examples of smooth maps between Euclidean spaces that are not diffeomorphisms

Techniques for Proving Smoothness

• Use the definition of smoothness in local coordinates and the properties of differentiable functions in Euclidean spaces
• Prove smoothness by showing that the map is differentiable and its differential is continuous
• Utilize the composition property of smooth maps to prove smoothness of composite functions
• Apply the inverse function theorem or the implicit function theorem to prove smoothness of locally defined maps
• Show that a map is a diffeomorphism by proving smoothness and constructing a smooth inverse
• Prove that a map is an immersion by computing the differential and showing injectivity
• Prove that a map is a submersion by computing the differential and showing surjectivity

Applications in Topology

• Smooth manifolds and smooth maps provide a foundation for studying differential topology and geometry
• Diffeomorphisms are used to define equivalence classes of smooth manifolds, leading to the notion of smooth structures
• Immersions and submersions are used to study the local and global properties of smooth maps between manifolds
• Morse theory studies the topology of smooth manifolds using critical points of smooth real-valued functions
  • Critical points are points where the differential of the function vanishes
• Degree theory for smooth maps between compact oriented manifolds of the same dimension provides a powerful topological invariant
  • Degree of a smooth map measures the number of preimages of a regular value, counted with signs
• Smooth structures on topological manifolds can be used to study exotic structures and the classification of manifolds
  • Exotic structures are smooth structures that are homeomorphic but not diffeomorphic to the standard smooth structure

Common Pitfalls and Misconceptions

• Not all continuous functions are smooth or even differentiable (Weierstrass function)
• Smoothness is a local property, while differentiability is a pointwise property
• Smoothness of a map depends on the smooth structures of the domain and codomain manifolds
• Diffeomorphisms preserve the smooth structure, but homeomorphisms may not
• Immersions and submersions are not necessarily injective or surjective as maps between sets
• Critical points of smooth maps are not always isolated or non-degenerate
• Smooth manifolds may admit different non-equivalent smooth structures (exotic spheres)

Practice Problems and Solutions

• Prove that the stereographic projection from the unit sphere minus the north pole to the Euclidean plane is a diffeomorphism
  • Solution: Show that the stereographic projection and its inverse are smooth maps using the definition of smoothness in local coordinates
• Determine the critical points of the height function on the torus and classify them as non-degenerate or degenerate
  • Solution: Parametrize the torus and compute the differential of the height function to find critical points, then compute the Hessian matrix at each critical point to determine non-degeneracy
• Prove that the Möbius strip is a smooth manifold and find a smooth immersion into Euclidean 3-space
  • Solution: Construct smooth transition functions between local coordinate charts of the Möbius strip, then use the parametrization of the Möbius strip as a ruled surface to define a smooth immersion
• Show that the unit sphere in Euclidean 3-space is diffeomorphic to the one-point compactification of the Euclidean plane
  • Solution: Construct a smooth map from the sphere to the one-point compactification using stereographic projection and prove that it is a diffeomorphism by showing smoothness of the inverse map
• Compute the degree of the antipodal map on the unit sphere and interpret the result topologically
  • Solution: Show that the antipodal map is smooth and its differential vanishes only at the poles, then compute the local degree at a regular value using the sign of the Jacobian determinant and conclude that the global degree is non-zero, implying that the antipodal map is not homotopic to a constant map