Differential Geometry

name: differential-geometry

description: > Expert differential geometry assistant for mathematicians and physicists. Use this skill whenever the user needs: help with smooth manifolds, Riemannian geometry, curvature, geodesics, or geometric structures on spaces. Includes both intrinsic differential geometry and applications to mathematical physics. trigger: Any problem involving smooth spaces, curvature, or geometric analysis - from pure mathematics to general relativity. license: MIT compatibility: opencode metadata: audience: mathematicians category: mathematics

Differential Geometry — Manifolds, Curvature, and Geometry

Covers: Smooth Manifolds · Tangent Spaces · Riemannian Metrics · Curvature · Geodesics · Connections · Lie Groups

Smooth Manifolds

Definition

An n-dimensional smooth (C^∞) manifold is a Hausdorff, second-countable space M with a smooth atlas: collection of charts (U_α, φ_α) where:

• U_α ⊂ M open cover
• φ_α: U_α → φ_α(U_α) ⊂ R^n
• Transition maps φ_β ∘ φ_α^{-1} are C^∞ where defined

Examples

Smooth Maps

A map f: M → N is smooth if in coordinates:

ψ ∘ f ∘ φ^{-1}: R^n → R^m

is C^∞ for all charts.

Diffeomorphism

Smooth map with smooth inverse. Two manifolds are diffeomorphic if such map exists.

Warning: Manifolds may be homeomorphic but not diffeomorphic (exotic spheres).

Submanifolds

N ⊂ M is embedded submanifold if inclusion is immersion and N has subspace topology.

Partition of Unity

For paracompact manifold, smooth partition of unity exists:

Σ_i φ_i = 1

Enables global constructions from local data.

Tangent Spaces

Tangent Vectors (Geometric)

Tangent vector at p ∈ M: equivalence class of curves γ(t) with γ(0) = p, tangent at t = 0.

Tangent Vectors (Algebraic)

Derivation at p: linear map v: C^∞(M) → R satisfying:

v(fg) = v(f)g(p) + f(p)v(g)

This gives intrinsic definition.

Tangent Space

Set of all tangent vectors at p:

T_pM

Dimension: dim T_pM = dim M

Coordinate Basis

If (x¹, ..., xⁿ) are local coordinates:

∂/∂x^i|_p forms basis of T_pM

Pushforward (Differential)

For smooth f: M → N:

f_*: T_pM → T_{f(p)}N

For curve: f_*(γ') = (f ∘ γ)'

Pullback of Covectors

For function f: M → R:

df_p: T_pM → R

Coordinate expression: df = (∂f/∂x^i) dx^i

Cotangent Space

Dual vector space:

T_p*M = (T_pM)*

Covectors: linear functionals on tangent vectors.

Exterior Product

Λ^k T_p*M = k-forms at p

Vector Bundles

Definition

Smooth fiber bundle with vector space fiber F = R^k and structure group GL(k, R).

Sections: smooth maps s: M → E with s(p) ∈ E_p.

Tangent Bundle

TM = ⊔_p T_pM

Section of TM: vector field on M.

Cotangent Bundle

T*M = ⊔_p T_p*M

Section: 1-form on M.

Line Bundle

Vector bundle of rank 1. Equivalent to divisor class (complex case).

Sections and Local Sections

Local section over U: s: U → E with π ∘ s = id_U.

Transition Functions

On overlap U_α ∩ U_β:

g_{αβ}: U_α ∩ U_β → GL(k,R)

Determines bundle from coordinate charts.

Operations on Bundles

• Dual: E* with fibers (E_p)*
• Tensor product: (E ⊗ F)_p = E_p ⊗ F_p
• Exterior power: Λ^k E
• Direct sum: E ⊕ F

Riemannian Metrics

Definition

Riemannian metric g: smooth assignment of inner product on tangent spaces:

g_p: T_pM × T_pM → R

Smoothly varying with p.

Length of Vector

|v|_p = √(g_p(v, v))

Length of Curve

For parameterized curve γ: [a,b] → M:

L(γ) = ∫_a^b |γ'(t)|_γ(t) dt

Riemannian Manifold

(M, g) with g a Riemannian metric.

Examples

• Euclidean space: g = δ_ij dx^i ⊗ dx^j
• Sphere S^n: induced metric from R^{n+1}
• Hyperbolic space: Poincaré ball model

Musical Isomorphisms

♭: TM → T*M, v ↦ g(v, ·)
♯: T*M → TM, ω ↦ g^{-1}(ω, ·)

Raising/lowering indices.

Norm and Angle

Angle between vectors:

cos θ = g(v,w)/(|v||w|)

Volume Form

For oriented Riemannian n-manifold:

dV_g = √|det g| dx¹ ∧ ... ∧ dx^n

Connections

Levi-Civita Connection

Unique connection ∇ on (M,g) satisfying:

1. Metric compatibility: X(g(Y,Z)) = g(∇_X Y, Z) + g(Y, ∇_X Z)
2. Torsion-free: ∇_X Y - ∇_Y X = [X,Y]

Christoffel symbols in coordinates:

∇_{∂/∂x^i} ∂/∂x^j = Γ^k_{ij} ∂/∂x^k

Covariant Derivative

For vector field Y along curve γ:

D_t Y = ∇_{γ'(t)} Y

This is the intrinsic derivative.

Parallel Transport

Y along γ is parallel if D_t Y = 0.

Given initial Y(0), parallel transport along γ gives Y(t).

Geodesics

Curve γ with γ'' = 0 (acceleration zero):

∇_{γ'} γ' = 0

In coordinates:

d²x^k/dt² + Γ^k_{ij} dx^i/dt dx^j/dt = 0

Exponential Map

For v ∈ T_pM, geodesic γ_v with γ_v(0) = p, γ_v'(0) = v.

Exponential map:

exp_p: U ⊂ T_pM → M, v → γ_v(1)

Local diffeomorphism near 0.

Logarithmic Map

Inverse of exp_p:

log_p: V ⊂ M → T_pM

Geodesic Distance

For Riemannian manifold:

d(p,q) = inf{L(γ) | γ from p to q}

For Minkowski space: d = straight line length.

Curvature

Riemann Curvature Tensor

R(X,Y)Z = ∇_X∇_Y Z - ∇_Y∇_X Z - ∇_{[X,Y]} Z

(1,3)-tensor. In coordinates: R^i_{jkl}.

Curvature Operator

R(X,Y): TM → TM, Z ↦ R(X,Y)Z

Sectional Curvature

For plane σ ⊂ T_pM:

K(σ) = g(R(e₁,e₂)e₂, e₁)/|e₁∧e₂|²

Depends only on the 2-plane.

Ricci Curvature

Trace of Riemann:

Ric(X,Y) = tr( Z → R(Z,X)Y )

In coordinates: R_{ij} = R^k_{ikj}.

Scalar Curvature

Trace of Ricci:

S = tr_g Ric = g^{ij} R_{ij}

Geometric Interpretations

• Sectional: curvature of 2D surface through plane
• Ricci: volume comparison, Einstein equations
• Scalar: average sectional curvature

Curvature and Topology

• Gauss-Bonnet: ∫_M K dA = 2π χ(M) (2D)
• Chern-Gauss-Bonnet: ∫_M χ(M) = (1/(8π)^{n/2} ∫ |P| (higher dimension)
• Bonnet-Myers: Ricci > (n-1)/R² → compact, diameter bounded

Comparison Geometry

• Rauch comparison: geodesic comparison based on curvature bounds
• Volume growth: lower bounds on Ricci → volume bounds
• Splitting theorem: Ricci ≥ 0 + line → product structure

Geodesics and Distance Geometry

Geodesic Equation

d²x^i/dt² + Γ^i_{jk} dx^j/dt dx^k/dt = 0

Second-order ODE → unique solution from initial conditions.

Geodesic Completeness

All geodesics extend for all time. Complete Riemannian manifolds: Hopf-Rinow theorem equivalent conditions.

Minimal Geodesics

Between p and q, minimizing length curve is geodesic (locally).

Cut Locus

Point where geodesic ceases to be minimizing. After cut point, another geodesic is shorter.

Injectivity Radius

inj(p) = min{distance to cut point in each direction}
inj(M) = inf_p inj(p)

Convexity

Geodesically convex: unique minimizing geodesic between any two points. Strictly convex: second fundamental form positive.

Length Minimization

• Hopf-Rinow: Complete + bounded → compact
• Gradient flow: geodesics as critical points of energy functional

Submanifolds

Immersions and Embeddings

f: N → M smooth with injective derivative.

Embedding: immersion + proper + topological embedding.

Induced Metric

For submanifold with immersion i: (N, i*g) Riemannian.

Second Fundamental Form

For submanifold Y ⊂ X:

II: T_pY × T_pY → (T_pY)^⊥

Measures extrinsic curvature.

Mean Curvature

Trace of second fundamental form:

H = tr(II)

Gauss Formula

For vector field along Y:

∇_X Y = (∇_X Y)^T + II(X,Y)

Weingarten Formula

For normal vector field:

∇_X ν = -A_ν(X) + ∇_X^⊥ ν

Where A_ν is shape operator.

Minimal Surfaces

Mean curvature H = 0 everywhere. Euler-Lagrange for area functional.

Lie Groups

Definition

Lie group G: smooth manifold with group structure (multiplication, inverse) smooth.

Examples

Lie Algebra

Tangent space at identity:

g = T_eG

Bracket: [X,Y] = XY - YX (commutator).

Exponential Map

For X ∈ g:

exp: g → G

For matrix groups: exp(X) = e^X.

One-Parameter Subgroups

γ(t) = exp(tX) satisfies:

• γ(0) = e
• γ'(0) = X
• γ(s+t) = γ(s)γ(t)

Adjoint Representation

Ad: G → Aut(g) Ad_g: X → gXg^{-1}

Derivative: ad: g → End(g), ad_X(Y) = [X,Y]

Homomorphisms

Lie group homomorphism φ: G → H gives Lie algebra homomorphism dφ: g → h.

Haar Measure

Unique left-invariant measure on Lie group. Integration using bi-invariant metric when available.

Curvature Computations

Curvature of Space Forms

Sphere S^n(R):

• Sectional curvature = 1/R²
• Ricci = (n-1)/R² g
• Scalar = n(n-1)/R²

Hyperbolic space H^n(R):

• Sectional curvature = -1/R²
• Ricci = -(n-1)/R² g
• Scalar = -n(n-1)/R²

Flat space:

• All curvatures = 0

Product Metrics

For (M₁,g₁) × (M₂,g₂):

• Curvature: sum of individual
• Ricci: diagonal sum
• Sectional: independent 2-planes

Warped Products

For f: M → (0,∞): M_f = M ×_f N with metric g = g_M + f² g_N.

Useful for cosmology, black holes.

Geometric Analysis

Laplace-Beltrami Operator

Δ_g f = div(∇f) = |g|^{-1/2} ∂_i(|g|^{1/2} g^{ij} ∂_j f)

Hodge Laplacian

Δ = dδ + δd on forms

Heat Kernel

Solution to heat equation:

(∂/∂t - Δ)K = 0
K(t,p,q) ~ (4πt)^{-n/2} e^{-d(p,q)²/4t}

Geodesic Flow

Flow on unit tangent bundle:

Φ_t: SM → SM

Anosov if sectional curvature < 0.

Geometric Evolution Equations

• Ricci flow: ∂g/∂t = -2Ric
• Mean curvature flow: ∂X/∂t = H
• Yamabe flow: ∂g/∂t = (R - r)g

Common Errors to Avoid

• Confusing tangent vectors with coordinate basis vectors
• Forgetting that curvature is a tensor (transforms correctly)
• Applying formulas without checking metric compatibility
• Confusing intrinsic and extrinsic curvature
• Forgetting that geodesics are locally length-minimizing, not globally
• Confusing the exponential map with matrix exponential (when different)
• Mixing up sectional, Ricci, and scalar curvature
• Not understanding that connection coefficients are not tensors
• Confusing pushforward and pullback directions
• Forgetting that Levi-Civita connection is metric compatible and torsion-free

Key References

• Riemannian Geometry by do Carmo — Standard intro
• Introduction to Riemannian Manifolds by Lee — Modern treatment
• Differential Geometry by Klingenberg — Advanced topics
• Foundations of Differential Geometry by Kobayashi & Nomizu — Comprehensive