Differential Geometry

Tangent Space

TₚM is the n-dimensional vector space of velocity vectors of curves through p — the linear approximation

The tangent space TₚM at a point p of an n-dimensional smooth manifold M is the n-dimensional vector space that "best approximates" M near p. Three equivalent definitions: (1) Equivalence classes of curves γ: (−ε, ε) → M with γ(0) = p, modulo the equivalence γ ~ δ iff (φ ∘ γ)'(0) = (φ ∘ δ)'(0) in any chart φ. (2) Derivations — linear maps D: C^∞(M) → ℝ satisfying the Leibniz rule. (3) In coordinates — the span of partial derivatives ∂/∂xⁱ at p. The disjoint union ⊔ TₚM is the tangent bundle TM, itself a 2n-manifold. Used in: vector fields (sections of TM), Lie groups (Lie algebra 𝔤 = T_e G), Riemannian metric (inner product on TₚM), and Einstein's general relativity (4D spacetime manifold with metric on tangent spaces).

  • Dimensiondim TₚM = dim M
  • Definitionscurves, derivations, coords
  • Basis∂/∂xⁱ
  • Tangent bundle TM2n-manifold
  • Lie algebraT_e of Lie group
  • GRT_p of spacetime

Watch the 60-second explainer

A condensed visual walkthrough — narrated, captioned, under a minute.

Why tangent space matters

  • Calculus on manifolds. The differential of a smooth map f: M → N at p is a linear map df_p: TₚM → T_{f(p)}N. Without tangent spaces, "derivative" on a curved space has no intrinsic meaning. The chain rule, inverse function theorem, and implicit function theorem all upgrade from ℝⁿ to manifolds via tangent-space linearization. Critical points and Morse theory live in the tangent space.
  • Lie groups and symmetry. Every Lie group G has a Lie algebra 𝔤 = T_e G with a bracket structure. The exponential map exp: 𝔤 → G turns infinitesimal symmetries (Lie algebra elements) into finite ones (group elements). Most of the structure of G is recoverable from 𝔤 — the local-to-global program of Lie theory — and computations in 𝔤 are linear-algebra computations.
  • General relativity. Spacetime is a 4-manifold M with a Lorentzian metric g. At each event p ∈ M, the tangent space TₚM is a 4-dimensional Minkowski space — locally the spacetime of special relativity. Light cones, timelike vectors, observers' velocities all live in TₚM. The Einstein field equations Gᵤᵥ = 8πG Tᵤᵥ live in T*ₚM ⊗ T*ₚM at each p. Curvature is built from second derivatives of the metric.
  • Mesh smoothing and computer graphics. Discrete tangent planes at vertices of a triangle mesh are the foundation of normal computation, Laplacian smoothing, and parameterization. In subdivision surfaces and mesh deformation, working with tangent vectors rather than ambient ℝ³ vectors gives intrinsic, embedding-independent algorithms.
  • Optimization on manifolds. Riemannian optimization (used in low-rank matrix completion, deep learning on the Stiefel manifold, sphere-constrained problems) requires the tangent-space formulation: gradient descent moves along TₚM, retraction maps back to M. Convergence and convexity are stated in terms of geodesic structure on TM.
  • Symplectic and Poisson geometry. The cotangent bundle T*M of any manifold is a canonical symplectic manifold. Hamiltonian mechanics — celestial mechanics, integrable systems, geometric quantization — is the theory of flows on T*M generated by Hamiltonian functions. Phase space is exactly cotangent space.

The three definitions in detail

Velocity classes

A tangent vector at p is an equivalence class of smooth curves γ: (−ε, ε) → M with γ(0) = p, where γ ~ δ if for some (equivalently every) chart φ around p, (φ ∘ γ)'(0) = (φ ∘ δ)'(0). The intuition: a tangent vector is "the velocity of a curve at p," and we identify curves whose first-order behaviour at p agrees. The vector-space structure is induced from ℝⁿ via any chart.

Derivations

A derivation at p is a linear map D: C^∞(M) → ℝ satisfying the Leibniz rule D(fg) = D(f) g(p) + f(p) D(g). The set of derivations forms a vector space — the algebraic tangent space at p. Given a curve γ as above, the operator D_γ(f) = (d/dt) f(γ(t)) |_{t=0} is a derivation. Hadamard's lemma shows every derivation arises this way; the bijection between curve classes and derivations is the canonical isomorphism between the geometric and algebraic definitions.

Coordinate basis

In a chart (U, φ) with coordinates x¹, …, xⁿ, the partial derivative operators ∂/∂x¹|_p, …, ∂/∂xⁿ|_p form a basis for TₚM as derivations: (∂/∂xⁱ|_p)(f) = ∂(f ∘ φ⁻¹)/∂xⁱ at φ(p). A tangent vector v has unique coordinates (a¹, …, aⁿ) with v = Σ aⁱ ∂/∂xⁱ|_p. Under a chart change, the components transform by the Jacobian — this is the contravariant tensor law that gave tangent vectors their classical name "contravariant vectors."

Worked examples

  • T_p ℝⁿ. Canonically isomorphic to ℝⁿ itself; the partial derivatives ∂/∂xⁱ act on functions in the obvious way.
  • T_p Sⁿ. The orthogonal complement of p ∈ Sⁿ ⊂ ℝⁿ⁺¹: TₚSⁿ = {v ∈ ℝⁿ⁺¹ : v · p = 0}, an n-dimensional subspace.
  • T_e SL(n, ℝ). Trace-zero matrices 𝔰𝔩(n, ℝ): the constraint det(g) = 1 linearizes at the identity to tr(X) = 0.
  • T_e SO(3). Skew-symmetric 3×3 matrices 𝔰𝔬(3) ≅ ℝ³ via X ↦ angular-velocity vector. Lie bracket = ordinary cross product.
  • T_p (M × N). T_p M × T_p N — tangent space of a product is the product of tangent spaces.
  • T_p (M ⊔ N). Whichever component p lies in; tangent spaces don't see other components.

Common misconceptions

  • Tangent space is part of the manifold. No. TₚM is a separate vector space attached to each point p. Identifying TₚM with a subset of M makes sense only when M is embedded in some ambient space and even then it is a coincidence of the embedding, not intrinsic. Two different points p, q have different tangent spaces TₚM and T_qM with no canonical isomorphism.
  • Only for manifolds in ℝⁿ. The intrinsic definitions (curves, derivations) make no reference to an ambient space. Abstract manifolds — Lie groups, projective spaces, moduli spaces — have tangent spaces defined intrinsically. Whitney's embedding theorem says abstract manifolds can be embedded in ℝ^{2n}, but this is a theorem, not a definition.
  • Always orthogonal to surface. "Orthogonal" requires an inner product; tangent space TₚM is just a vector space until a metric is chosen. For an embedded surface in ℝ³, the tangent plane is orthogonal to the surface normal in the ambient inner product, but for an abstract Lie group there is no ambient space and "orthogonality" is meaningless without a chosen Riemannian metric.
  • TₚM and T*ₚM are the same. They have the same dimension, but no canonical isomorphism without extra structure. A Riemannian metric provides the musical isomorphism (lowering and raising indices) between TₚM and T*ₚM. Tangent vectors transform contravariantly; cotangent vectors (1-forms) transform covariantly. In physics, "vector" usually means tangent (velocity-like) and "covector" means cotangent (gradient-like).
  • TM is a product M × ℝⁿ. Locally yes (in any chart, TU ≅ U × ℝⁿ), but globally usually not. TS² is not S² × ℝ²; the hairy ball theorem says no global trivialization exists. Manifolds whose tangent bundle is trivial are called parallelizable; among spheres only S¹, S³, S⁷ are parallelizable.
  • Tangent vectors equal directional-derivative formulas. The directional derivative is the action of a tangent vector on a function — the result of evaluating v(f). The vector v itself is the operator (or curve class), not the resulting number. Conflating them is harmless in coordinates but confuses the geometry.

Frequently asked questions

What is a tangent vector physically (velocity)?

A tangent vector at p is the velocity of a smooth curve through p — the instantaneous direction and speed of motion. Concretely, take a smooth γ: (−ε, ε) → M with γ(0) = p; its velocity γ'(0) is a tangent vector at p. Two curves represent the same tangent vector if their velocities agree in some (equivalently every) coordinate chart. Physically: a particle's instantaneous velocity in a curved space lives in the tangent space at its current location, not in any ambient Euclidean space.

Why are derivations equivalent to velocity classes?

Given a curve γ at p, the operation D_γ(f) = (d/dt) f(γ(t)) at t = 0 is linear in f and satisfies the Leibniz rule D_γ(fg) = D_γ(f) g(p) + f(p) D_γ(g) — that is, D_γ is a derivation at p. Conversely, every derivation arises this way: by Hadamard's lemma, any smooth function near p decomposes as f(p) + Σxⁱ gᵢ(x), so a derivation is determined by its values on the coordinate functions, which form a vector aⁱ ∈ ℝⁿ; the curve γ(t) = p + ta in coordinates realizes the derivation. The bijection makes the geometric and algebraic definitions equivalent.

What is the tangent bundle?

The tangent bundle TM is the disjoint union of all tangent spaces TₚM as p ranges over M, given a smooth manifold structure compatible with the projection π: TM → M, π(v ∈ TₚM) = p. Local trivialization: in any chart U on M, TU ≅ U × ℝⁿ. TM is itself a smooth 2n-manifold. Sections of TM are vector fields. TS² is non-trivial (no global trivialization, by hairy ball); TS¹ ≅ S¹ × ℝ is trivial; TS³ is trivial (S³ is a Lie group). Whitney embedding: TM embeds in ℝ^{2n+1} for compact M.

How does the Lie algebra arise as a tangent space?

A Lie group G is a smooth manifold that is also a group with smooth multiplication and inversion. The tangent space at the identity, T_e G, has a special structure: left-invariant vector fields on G correspond bijectively to elements of T_e G, and the Lie bracket of vector fields restricts to a bracket [·, ·]: T_e G × T_e G → T_e G satisfying antisymmetry and Jacobi. T_e G with this bracket is the Lie algebra 𝔤. Examples: 𝔤𝔩(n, ℝ) = T_I GL(n, ℝ) = M_n(ℝ) with [A, B] = AB − BA; 𝔰𝔬(3) = skew-symmetric 3×3 matrices, the angular-velocity space. Linearized Lie group structure encoded in TₑG.

What is a Riemannian metric and how does it use TₚM?

A Riemannian metric g on M is a smoothly varying choice of inner product gₚ: TₚM × TₚM → ℝ for each p. With g, lengths of tangent vectors are defined (|v|² = gₚ(v, v)), so curves have length ∫|γ'(t)| dt and the manifold has a distance function. Geodesics are length-minimizing curves; sectional curvature K(σ) is a function of 2-planes σ ⊂ TₚM. Einstein's GR uses a pseudo-Riemannian metric (signature (−, +, +, +)) on a 4D spacetime manifold; gₚ measures spacetime intervals at each event p, and the Einstein field equations relate curvature to stress-energy.

How is the cotangent space TₚM* used (1-forms)?

The cotangent space TₚM* is the dual vector space of TₚM — linear functionals on tangent vectors. Differential forms ω evaluate to ωₚ ∈ TₚM* at each p; the most basic 1-forms are exterior derivatives df where f: M → ℝ is smooth. In coordinates, df = Σ (∂f/∂xⁱ) dxⁱ where {dxⁱ} is the basis of TₚM* dual to {∂/∂xⁱ}. Cotangent bundle T*M is the natural setting for Hamiltonian mechanics: positions in M, momenta in T*M; the symplectic form ω = dpᵢ ∧ dqⁱ makes T*M into a symplectic manifold, and Hamilton's equations are Hamiltonian flow on T*M.