Lesson 5: Differential Geometry I

Part 4: Riemannian Metrics and Connections

Measuring and Comparing in Curved Space

When we walk on the surface of the Earth, we can measure distances along our path and determine angles between different directions. These measurements are possible because the Earth’s surface has a metric structure. Similarly, to do physics on any manifold, we need ways to measure distances and angles, and to compare vectors at different points. This is where metrics and connections come in.

Let’s organize our exploration using the mnemonic “GEOMETRY”:

The Riemannian Metric

A Riemannian metric g is a symmetric, positive-definite tensor field that measures distances and angles. At each point p, it provides an inner product on the tangent space TpM:

1. Definition and Properties For vectors v, w ∈ TpM: $g(v,w) = g_{ij}v^iw^j$

Requirements:

• Symmetry: g(v,w) = g(w,v)

• Positive definiteness: g(v,v) > 0 for v ≠ 0

• Smoothness: gᵢⱼ are smooth functions

2. Length and Distance For a curve γ(t): $L[\gamma] = \int_a^b \sqrt{g_{ij}\frac{dx^i}{dt}\frac{dx^j}{dt}}dt$

Distance between points: $d(p,q) = \inf_{\gamma} > L[\gamma]$ where γ runs over curves from p to q.

3. Geometric Quantities

   • Angles: $\cos\theta = \frac{g(v,w)}{\sqrt{g(v,v)g(w,w)}}$

   • Volume element: $dV = \sqrt{|g|}dx^1\wedge...\wedge dx^n$

   • Area of parallelogram: $A = \sqrt{g(v,v)g(w,w) - g(v,w)^2}$

Connections and Parallel Transport

A connection ∇ tells us how to compare vectors at different points, essentially defining “parallel transport” along curves:

1. Formal Definition ∇: X(M) × X(M) → X(M) satisfying:

   • Linearity: ∇ₐX+bY Z = a∇ₓZ + b∇ᵧZ

   • Leibniz rule: ∇ₓ(fY) = X(f)Y + f∇ₓY

Local expression: $\nabla_{\partial_i}\partial_j = > \Gamma^k_{ij}\partial_k$

where Γᵏᵢⱼ are Christoffel symbols.

2. Christoffel Symbols For the Levi-Civita connection:

   $$\Gamma^k_{ij} = \frac{1}{2}g^{kl}(\partial_ig_{jl} + \partial_jg_{il} - \partial_lg_{ij})$$

Properties:

• Symmetric in lower indices

• Determined by metric compatibility

• Transform non-tensorially

3. Parallel Transport Along curve γ(t): $\frac{DV^i}{dt} = \frac{dV^i}{dt} + \Gamma^i_{jk}\frac{dx^j}{dt}V^k = 0$

This equation defines parallel transport.

Geodesics: Nature’s Straightest Paths

Geodesics are paths that parallel transport their own tangent vectors:

1. Geodesic Equation $\frac{d^2x^i}{dt^2} + \Gamma^i_{jk}\frac{dx^j}{dt}\frac{dx^k}{dt} = 0$

This is equivalent to: $\nabla_{\dot{\gamma}}\dot{\gamma} = > 0$

2. Properties

   • Locally minimize distance

   • Preserve speed: g(γ̇,γ̇) constant

   • Generalize straight lines

Physical significance:

• Free particle motion

• Light rays in GR

• Minimal action paths

Curvature: Measuring Geometric Deviation

Curvature measures how parallel transport depends on path:

1. Riemann Tensor $R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z$

Components: $R^i_{jkl} = \partial_k\Gamma^i_{jl} - > \partial_l\Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - > \Gamma^m_{jk}\Gamma^i_{ml}$

2. Derived Quantities

   • Ricci tensor: Rᵢⱼ = Rᵏᵢₖⱼ

   • Scalar curvature: R = gⁱʲRᵢⱼ

   • Weyl tensor: Conformal part of Riemann

3. Geometric Meaning

   • Measures non-commutativity of covariant derivatives

   • Quantifies deviation from flat space

   • Controls geodesic deviation

Physical Applications

1. General Relativity Einstein’s equations: $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$

Shows how matter curves spacetime.

2. Geodesic Motion Test particles follow geodesics:

   $$\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\nu\lambda}\frac{dx^\nu}{d\tau}\frac{dx^\lambda}{d\tau} = 0$$

This is Einstein’s principle of inertia.

3. Yang-Mills Theory Connection as gauge field: $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu,A_\nu]$

Curvature as field strength.

Advanced Topics

1. Holonomy Groups Parallel transport around loops:

   • Measures global geometry

   • Classifies special geometries

   • Connects to gauge theory

2. Killing Fields Vector fields preserving metric: $\mathcal{L}_Xg = 0$

Generate isometries.

3. Sectional Curvature $K(π) = \frac{R(X,Y,X,Y)}{g(X,X)g(Y,Y) - g(X,Y)^2}$

Measures curvature in 2-planes.

Practical Calculations

1. Spherical Metric On S²: $ds^2 = R^2(d\theta^2 + \sin^2\theta d\phi^2)$

Calculate:

• Christoffel symbols

• Geodesic equations

• Riemann tensor

2. Schwarzschild Solution Metric: $ds^2 = -(1-\frac{2M}{r})dt^2 + (1-\frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$

Study:

• Geodesics

• Event horizon

• Curvature singularity

Exercises for Understanding

1. Basic Computation For the metric ds² = dx² + e²ˣdy²:

   • Find Christoffel symbols

   • Write geodesic equations

   • Calculate scalar curvature

2. Physical Application Study null geodesics in Schwarzschild spacetime:

   • Write equations of motion

   • Find circular orbits

   • Calculate bending of light

3. Geometric Understanding Prove that any two-dimensional metric is locally conformally flat: $g_{ij} = e^{2f}\delta_{ij}$

Looking Ahead

In our next section, we’ll explore how these geometric concepts extend to complex manifolds and Kähler geometry. We’ll see how:

• Complex structure enriches geometry

• Hermitian metrics arise

• Holomorphic functions behave

Remember: Metrics and connections aren’t just mathematical structures — they tell us how space itself is shaped and how things move through it. Understanding them deeply means understanding the stage on which physics plays out.

Next: Part 5 - Complex Manifolds and Kähler Geometry