📐Metric Differential Geometry Unit 7 – Connections and Parallel Transport

Key Concepts and Definitions

• Manifold: A topological space that locally resembles Euclidean space near each point
• Tangent space: A vector space attached to each point of a manifold, consisting of all possible directions in which one can tangentially pass through that point
• Connection: A way of defining parallel transport and the covariant derivative on a manifold
  • Allows for the comparison of vectors and tensors at different points on the manifold
  • Specifies how to transport vectors along curves while preserving their direction
• Parallel transport: The process of moving a vector along a curve on a manifold while keeping it "parallel" according to the connection
• Covariant derivative: An extension of the ordinary derivative that takes into account the change in the coordinate system as one moves along the manifold
• Curvature: A measure of how much the manifold deviates from being flat (Euclidean)
  • Determined by the Riemann curvature tensor, which quantifies the non-commutativity of parallel transport around infinitesimal loops
• Torsion: A measure of the failure of the connection to be symmetric, i.e., the difference between the covariant derivatives taken in different orders

Geometric Intuition

• Manifolds can be thought of as curved spaces that locally resemble Euclidean space (e.g., the surface of a sphere or a torus)
• Tangent spaces provide a way to describe vectors and directions at each point on the manifold
  • Tangent vectors can be visualized as arrows attached to points on the manifold
• Connections define a notion of parallelism between tangent spaces at different points
  • Imagine sliding a vector along a curve on the manifold while keeping it "straight" according to the connection
• Parallel transport can be visualized as moving a vector along a curve without changing its direction relative to the connection
  • On a sphere, parallel transport along a closed curve (e.g., a great circle) results in the vector pointing in a different direction when it returns to the starting point
• Curvature measures how much the manifold deviates from being flat
  • A sphere has positive curvature, while a saddle-shaped surface has negative curvature
  • Flat spaces (e.g., a plane or a cylinder) have zero curvature
• Torsion can be understood as a twisting of the manifold, causing vectors to rotate as they are parallel transported along certain paths

Connection on a Manifold

• A connection is a mathematical object that defines parallel transport and the covariant derivative on a manifold

• Connections are typically specified by a set of connection coefficients (Christoffel symbols) $\Gamma^i_{jk}$, which determine how the basis vectors of the tangent space change as one moves along the manifold

  • Connection coefficients are not tensors and depend on the choice of coordinate system

• The covariant derivative $\nabla_X Y$ of a vector field $Y$ in the direction of a vector field $X$ is defined using the connection coefficients:

  $\nabla_X Y = X^i \left(\frac{\partial Y^j}{\partial x^i} + \Gamma^j_{ik} Y^k\right) \frac{\partial}{\partial x^j}$

• Connections can be classified into various types:

  • Levi-Civita connection: A unique, torsion-free connection compatible with the metric (preserves inner products under parallel transport)
  • Metric connections: Connections that are compatible with a given metric on the manifold
  • Flat connections: Connections with zero curvature, allowing for global parallel transport without path dependence

• The choice of connection depends on the geometric properties and symmetries of the manifold and the specific problem at hand

Parallel Transport

• Parallel transport is the process of moving a vector along a curve on a manifold while keeping it "parallel" according to the connection
• Given a curve $\gamma(t)$ and a vector $v$ at the starting point $\gamma(0)$, parallel transport defines a unique vector field $V(t)$ along the curve such that:
  • $V(0) = v$
  • $\nabla_{\dot{\gamma}(t)} V(t) = 0$ (the covariant derivative of $V(t)$ along the curve is zero)
• Parallel transport is path-dependent in general, meaning that the resulting vector at the endpoint of the curve depends on the specific path taken
  • This path-dependence is a consequence of curvature
  • In flat spaces or spaces with a flat connection, parallel transport is path-independent
• Parallel transport can be used to compare vectors and tensors at different points on the manifold
  • This is essential for defining geometric quantities such as curvature and for formulating physical theories on curved spaces
• The holonomy group of a connection is the group of linear transformations obtained by parallel transporting vectors along closed loops starting and ending at a given point
  • The holonomy group provides information about the global geometric properties of the manifold and the connection

Curvature and Torsion

• Curvature is a measure of how much the manifold deviates from being flat (Euclidean)
• The Riemann curvature tensor $R^i_{jkl}$ quantifies the curvature of a manifold with a given connection

  • It measures the non-commutativity of parallel transport around infinitesimal loops

  • The components of the Riemann tensor can be expressed in terms of the connection coefficients and their derivatives:

    $R^i_{jkl} = \frac{\partial \Gamma^i_{jl}}{\partial x^k} - \frac{\partial \Gamma^i_{jk}}{\partial x^l} + \Gamma^i_{mk} \Gamma^m_{jl} - \Gamma^i_{ml} \Gamma^m_{jk}$

• The Ricci tensor $R_{ij}$ and scalar curvature $R$ are contractions of the Riemann tensor:
  • Ricci tensor: $R_{ij} = R^k_{ikj}$
  • Scalar curvature: $R = g^{ij} R_{ij}$ (where $g^{ij}$ is the inverse metric tensor)
• Torsion is a measure of the failure of the connection to be symmetric

  • The torsion tensor $T^i_{jk}$ is defined as the antisymmetric part of the connection coefficients:

    $T^i_{jk} = \Gamma^i_{jk} - \Gamma^i_{kj}$

• A connection is called torsion-free if its torsion tensor vanishes identically
  • The Levi-Civita connection is always torsion-free
• Curvature and torsion provide essential information about the geometric properties of the manifold and the connection
  • They play a crucial role in the formulation of physical theories, such as general relativity and gauge theories

Applications in Physics

• General relativity: Einstein's theory of gravity, which describes spacetime as a 4-dimensional manifold with a Lorentzian metric
  • The curvature of spacetime is determined by the presence of matter and energy, as described by the Einstein field equations
  • Parallel transport and geodesics (curves that parallel transport their own tangent vector) describe the motion of particles and light in curved spacetime
• Gauge theories: Mathematical frameworks that describe the fundamental interactions of particles in terms of connections on principal bundles
  • Examples include electromagnetism, the weak interaction, and the strong interaction
  • The curvature of the connection corresponds to the field strength (e.g., the electromagnetic field tensor)
  • Parallel transport describes the evolution of particle states under the influence of the gauge fields
• Geometric mechanics: The study of classical mechanical systems using differential geometry
  • The configuration space of a mechanical system can be described as a manifold, with the dynamics determined by a connection (e.g., the Levi-Civita connection for a Riemannian metric)
  • Parallel transport and curvature provide insights into the conservation laws and symmetries of the system
• Geometric phases: The phase factors acquired by quantum states when they undergo cyclic evolution in parameter space
  • Examples include the Berry phase and the Aharonov-Bohm effect
  • Geometric phases can be understood in terms of parallel transport and holonomy in a suitable vector bundle over the parameter space

Examples and Calculations

• Parallel transport on a sphere:
  • Consider a unit sphere $S^2$ with the standard metric induced from $\mathbb{R}^3$
  • The Levi-Civita connection on the sphere is determined by the requirement that the tangent vectors to great circles are parallel transported along the great circles
  • Parallel transporting a vector along a closed path (e.g., a great circle) results in a rotation of the vector, demonstrating the path-dependence of parallel transport
• Curvature of a torus:

  • A torus can be parametrized by two angles $(\theta, \phi)$, with the metric given by:

    $ds^2 = (R + r \cos\theta)^2 d\phi^2 + r^2 d\theta^2$

  • The Riemann curvature tensor can be calculated using the Christoffel symbols derived from the metric

  • The scalar curvature of the torus is non-constant and depends on the radii $R$ and $r$

• Parallel transport in a gauge theory:
  • Consider a U(1) gauge theory (electromagnetism) on a spacetime manifold
  • The gauge connection is given by the electromagnetic potential $A_\mu$, and the curvature corresponds to the electromagnetic field tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$
  • Parallel transport of a charged particle's wavefunction along a closed path in spacetime leads to a phase factor (the Aharonov-Bohm effect) determined by the holonomy of the connection

Advanced Topics and Extensions

• Fiber bundles: A generalization of manifolds that includes additional structure, such as vector bundles and principal bundles
  • Connections on fiber bundles provide a unified framework for describing gauge theories and geometric phases
• Characteristic classes: Topological invariants associated with vector bundles and principal bundles, which provide information about the global structure of the bundle and its connection
  • Examples include the Chern classes, the Pontryagin classes, and the Euler class
• Holonomy and the Ambrose-Singer theorem: A deep result relating the curvature of a connection to its holonomy group
  • The theorem states that the Lie algebra of the holonomy group is generated by the curvature tensors evaluated at all points of the manifold
• Symplectic and Poisson manifolds: Manifolds equipped with a symplectic form or a Poisson bracket, which provide a geometric framework for classical mechanics and Hamiltonian systems
  • Connections on symplectic and Poisson manifolds (e.g., symplectic connections and Poisson connections) play a role in the geometric formulation of mechanics and field theories
• Kähler manifolds: Complex manifolds with a compatible Riemannian metric and a symplectic form
  • The Levi-Civita connection on a Kähler manifold has special properties, such as the preservation of the complex structure under parallel transport
  • Kähler manifolds are important in complex geometry, algebraic geometry, and string theory
• Generalized geometry: An approach that unifies symplectic and complex geometry by considering structures on the direct sum of the tangent and cotangent bundles
  • Connections in generalized geometry (e.g., Courant algebroids and generalized complex structures) provide a framework for studying T-duality and other aspects of string theory