🔁Elementary Differential Topology Unit 3 – Tangent Spaces and Map Differentials

Key Concepts and Definitions

• Tangent space $T_pM$ the set of all tangent vectors to a manifold $M$ at a point $p$
• Tangent vector an equivalence class of curves on a manifold that agree to first order at a point
  • Represents a direction and magnitude of instantaneous change at a point on a manifold
• Map differential $(df)_p: T_pM \to T_{f(p)}N$ a linear map between tangent spaces induced by a smooth map $f: M \to N$
  • Describes how tangent vectors are transformed by the map $f$ at each point
• Pushforward $(f_*)_p: T_pM \to T_{f(p)}N$ another notation for the map differential, emphasizing the "pushing forward" of tangent vectors
• Pullback $f^*: T^*_{f(p)}N \to T^*_pM$ the dual map to the map differential, acting on cotangent spaces (dual spaces to tangent spaces)
• Tangent bundle $TM$ the disjoint union of all tangent spaces of a manifold $M$, forming a new manifold
• Local coordinates $(x^1, \dots, x^n)$ a system of coordinate functions on a manifold that provides a local parameterization around each point

Tangent Spaces: The Basics

• Tangent spaces provide a linear approximation to a manifold at each point
  • Allows for calculus and linear algebra techniques to be applied to manifolds
• Each tangent space $T_pM$ is a vector space of the same dimension as the manifold $M$
• Tangent vectors can be represented as derivations on the algebra of smooth functions at a point
  • $v(f) = (v^1 \frac{\partial}{\partial x^1} + \dots + v^n \frac{\partial}{\partial x^n})(f)$ in local coordinates
• The tangent space at a point is independent of the choice of local coordinates
  • Coordinate changes induce linear isomorphisms between tangent spaces
• Tangent vectors can be visualized as arrows attached to points on the manifold
  • The set of all tangent vectors forms a "fuzzy ball" around each point
• Smooth curves on the manifold correspond to smooth vector fields (sections of the tangent bundle)
  • The tangent vector to a curve at a point is the velocity vector of the curve at that point

Constructing Tangent Spaces

• Tangent spaces can be constructed using equivalence classes of curves
  • Two curves $\gamma_1, \gamma_2: (-\epsilon, \epsilon) \to M$ are equivalent at $p = \gamma_1(0) = \gamma_2(0)$ if they have the same derivative at $0$ in any local coordinate system
  • The equivalence class $[\gamma]_p$ is a tangent vector at $p$
• Alternatively, tangent vectors can be defined as derivations on the algebra of germs of smooth functions at a point
  • A derivation $v: C^\infty_p(M) \to \mathbb{R}$ satisfies the Leibniz rule: $v(fg) = v(f)g(p) + f(p)v(g)$
• The tangent space $T_pM$ is the set of all derivations at $p$, which forms a vector space
  • The dimension of $T_pM$ equals the dimension of $M$
• In local coordinates $(x^1, \dots, x^n)$, the partial derivatives $\frac{\partial}{\partial x^i}|_p$ form a basis for $T_pM$
  • Any tangent vector can be expressed as a linear combination $v = v^i \frac{\partial}{\partial x^i}|_p$
• The dual space to the tangent space is the cotangent space $T^*_pM$, consisting of linear functionals on tangent vectors
  • The differentials $dx^i|_p$ of the coordinate functions form a basis for $T^*_pM$

Map Differentials Explained

• Given a smooth map $f: M \to N$ between manifolds, the map differential $(df)_p: T_pM \to T_{f(p)}N$ is a linear map between tangent spaces
  • Intuitively, $(df)_p$ describes how $f$ transforms tangent vectors from $M$ to $N$
• The map differential is defined by $(df)_p([\gamma]_p) = [f \circ \gamma]_{f(p)}$ for curves $\gamma$ in $M$
  • Equivalently, $(df)_p(v)(g) = v(g \circ f)$ for derivations $v$ and functions $g$
• In local coordinates $(x^i)$ on $M$ and $(y^j)$ on $N$, the map differential is represented by the Jacobian matrix $(\frac{\partial f^j}{\partial x^i})$
  • $(df)_p(\frac{\partial}{\partial x^i}|_p) = \frac{\partial f^j}{\partial x^i}(p) \frac{\partial}{\partial y^j}|_{f(p)}$
• The map differential satisfies the chain rule: if $f: M \to N$ and $g: N \to P$, then $d(g \circ f)_p = (dg)_{f(p)} \circ (df)_p$
• The map differential is a functor from the category of smooth manifolds to the category of vector spaces
  • It preserves composition and identities: $d(\mathrm{id}_M)_p = \mathrm{id}_{T_pM}$ and $d(g \circ f)_p = (dg)_{f(p)} \circ (df)_p$
• The pullback $f^*: T^*_{f(p)}N \to T^*_pM$ is defined by $\langle f^*(\omega), v \rangle = \langle \omega, (df)_p(v) \rangle$ for $\omega \in T^*_{f(p)}N$ and $v \in T_pM$
  • In local coordinates, $f^*(dy^j|_{f(p)}) = \frac{\partial f^j}{\partial x^i}(p) dx^i|_p$

Applications in Differential Topology

• Tangent spaces and map differentials are fundamental tools in differential topology
• The rank of a smooth map $f: M \to N$ at a point $p$ is defined as the rank of the linear map $(df)_p$
  • A point $p$ is a regular point if $(df)_p$ has maximal rank (equal to $\min(\dim M, \dim N)$)
  • The set of regular points is open and dense in $M$ (Sard's theorem)
• Submersions are maps $f: M \to N$ with $(df)_p$ surjective at each point
  • The preimage $f^{-1}(q)$ of a regular value $q \in N$ is a submanifold of $M$ of dimension $\dim M - \dim N$
• Immersions are maps $f: M \to N$ with $(df)_p$ injective at each point
  • The image $f(M)$ is locally a submanifold of $N$ of dimension $\dim M$
• Embeddings are immersions that are also homeomorphisms onto their image
  • Provide a way to realize abstract manifolds as submanifolds of Euclidean space
• The tangent bundle $TM$ is a manifold of dimension $2\dim M$, with a natural projection $\pi: TM \to M$
  • Vector fields on $M$ correspond to sections of the tangent bundle (smooth maps $X: M \to TM$ with $\pi \circ X = \mathrm{id}_M$)
• Riemannian metrics on a manifold are smooth choices of inner products on each tangent space
  • Allow for the definition of lengths, angles, and geodesics on the manifold

Common Techniques and Problem-Solving

• Computing map differentials in local coordinates using the Jacobian matrix
  • Expressing tangent vectors in terms of coordinate bases and applying the Jacobian
• Determining the rank of a map differential at a point by computing the rank of the Jacobian matrix
  • Using the rank to classify points as regular or critical
• Finding the preimages of regular values using the implicit function theorem
  • Showing that the preimage is a submanifold of the appropriate dimension
• Constructing embeddings of manifolds into Euclidean space using the Whitney embedding theorem
  • Showing that a map is an immersion and a homeomorphism onto its image
• Solving differential equations on manifolds using vector fields and flows
  • Expressing the differential equation in local coordinates and using existence and uniqueness theorems
• Analyzing the behavior of geodesics on Riemannian manifolds using the exponential map
  • Computing Christoffel symbols and solving the geodesic equation in local coordinates
• Proving topological properties of manifolds using transversality arguments
  • Showing that certain maps can be perturbed to be transverse to submanifolds, and using the preimage theorem

Connections to Other Topics

• Tangent spaces and map differentials are closely related to the concept of derivatives in multivariable calculus
  • The Jacobian matrix of a map between Euclidean spaces is a special case of the map differential
• The tangent bundle is an example of a vector bundle, a fundamental object in algebraic topology
  • Other examples include the cotangent bundle, normal bundle, and tensor bundles
• Riemannian geometry heavily relies on the theory of tangent spaces and map differentials
  • Riemannian metrics, connections, and curvature are defined using these concepts
• Lie groups are manifolds equipped with a group structure, where the group operations are smooth maps
  • The tangent space at the identity of a Lie group is called the Lie algebra, which captures the infinitesimal structure of the group
• Symplectic geometry studies manifolds with a closed, nondegenerate 2-form (symplectic form)
  • Symplectic manifolds have a rich structure related to Hamiltonian mechanics and geometric quantization
• Morse theory analyzes the topology of manifolds using the critical points of smooth functions
  • The index of a critical point is defined using the Hessian matrix, which is related to the second derivative of the function
• Characteristic classes are cohomological invariants associated with vector bundles, often defined using the map differentials of classifying maps
  • Examples include Chern classes, Pontryagin classes, and Euler classes

Tricky Points and Common Mistakes

• Confusing the tangent space $T_pM$ at a point with the tangent bundle $TM$ over the entire manifold
  • The tangent space is a vector space, while the tangent bundle is a manifold
• Incorrectly applying the chain rule for map differentials
  • The order of composition matters: $d(g \circ f)_p = (dg)_{f(p)} \circ (df)_p$, not $(df)_p \circ (dg)_{f(p)}$
• Forgetting to check the smoothness of maps when working with map differentials
  • The map differential is only defined for smooth maps between manifolds
• Misinterpreting the rank of a map differential
  • A map can have different ranks at different points, and the rank can change even if the map is smooth
• Misapplying the preimage theorem for submersions
  • The preimage of a regular value is a submanifold, but the preimage of a critical value may not be
• Confusing immersions and embeddings
  • An immersion need not be injective, while an embedding must be injective and a homeomorphism onto its image
• Incorrectly computing Christoffel symbols or geodesic equations in local coordinates
  • The formulas involve partial derivatives of the metric coefficients and can be tricky to manipulate
• Misunderstanding the relationship between tangent vectors and derivations
  • Tangent vectors can be represented as derivations, but not all derivations correspond to tangent vectors (e.g., higher-order derivations)