Affine connections are the backbone of differential geometry, allowing us to compare vectors at different points on a manifold. They're crucial for understanding how things move and change in curved spaces, like in general relativity.

Connections give us tools like covariant derivatives and parallel transport. These help us study how vectors and tensors change along curves, leading to important concepts like geodesics and curvature. It's all about measuring geometry without relying on coordinates.

Affine Connections and Christoffel Symbols

Understanding Affine Connections

• Affine connection defines parallel transport on a manifold, allowing comparison of vectors at different points
• Provides a way to differentiate vector fields on a manifold, extending the concept of directional derivatives
• Assigns to each pair of vector fields X and Y a new vector field $\nabla_X Y$, called the covariant derivative of Y along X
• Satisfies linearity and Leibniz rule properties:
  • Linearity in X: $\nabla_{aX + bY} Z = a\nabla_X Z + b\nabla_Y Z$ for scalar functions a and b
  • Linearity in Y: $\nabla_X (aY + bZ) = a\nabla_X Y + b\nabla_X Z$ for scalar functions a and b
  • Leibniz rule: $\nabla_X (fY) = (Xf)Y + f\nabla_X Y$ for a scalar function f
• Enables the study of geometric properties independent of coordinate systems

Connection Coefficients and Christoffel Symbols

• Connection coefficients represent components of the affine connection in a local coordinate system
• Expressed as $\Gamma^k_{ij}$ in a coordinate basis, where i, j, and k are indices
• Christoffel symbols serve as specific connection coefficients for the Levi-Civita connection
• Calculated using the metric tensor g and its derivatives: $\Gamma^k_{ij} = \frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$
• Transformation law for connection coefficients under coordinate change: $\Gamma'^k_{ij} = \frac{\partial x'^k}{\partial x^l} \Gamma^l_{mn} \frac{\partial x^m}{\partial x'^i} \frac{\partial x^n}{\partial x'^j} + \frac{\partial x'^k}{\partial x^l} \frac{\partial^2 x^l}{\partial x'^i \partial x'^j}$

Covariant Derivative and Its Applications

• Covariant derivative extends the notion of directional derivative to curved spaces
• Defined for vector fields X and Y as $\nabla_X Y = X^i (\partial_i Y^j + \Gamma^j_{ik} Y^k) \partial_j$
• Measures how a vector field changes along a curve on the manifold
• Used to define parallel transport: a vector field Y is parallel along a curve γ if $\nabla_{\dot{\gamma}} Y = 0$
• Generalizes to tensors of arbitrary rank, preserving tensor structure
• Applications include:
  • Geodesic equations: $\frac{d^2 x^i}{dt^2} + \Gamma^i_{jk} \frac{dx^j}{dt} \frac{dx^k}{dt} = 0$
  • Parallel transport of vectors along curves
  • Definition of curvature and torsion tensors

Torsion and Curvature

Torsion Tensor: Measuring Non-Commutativity

• Torsion tensor T measures the failure of the affine connection to be symmetric
• Defined as $T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$ for vector fields X and Y
• Components in a coordinate basis: $T^i_{jk} = \Gamma^i_{jk} - \Gamma^i_{kj}$
• Vanishes for the Levi-Civita connection, making it torsion-free
• Non-zero torsion indicates:
  • Path dependence in parallel transport
  • Non-closure of infinitesimal parallelograms
• Plays a role in theories of gravity with spin (Einstein-Cartan theory)

Curvature Tensor: Measuring Intrinsic Geometry

• Curvature tensor R measures the failure of parallel transport to be path-independent
• Defined as $R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$ for vector fields X, Y, and Z
• Components in a coordinate basis (Riemann tensor): $R^i_{jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^i_{km} \Gamma^m_{jl} - \Gamma^i_{lm} \Gamma^m_{jk}$
• Measures how much the geometry deviates from Euclidean space
• Properties include:
  • Antisymmetry in last two indices: $R^i_{jkl} = -R^i_{jlk}$
  • Bianchi identities: $R^i_{j[kl;m]} = 0$ and $R^i_{j[kl;m]} + T^n_{[kl}R^i_{|j|m]n} = 0$
• Used to define:
  • Ricci tensor: $R_{ij} = R^k_{ikj}$
  • Scalar curvature: $R = g^{ij}R_{ij}$
• Applications in general relativity and differential geometry:
  • Einstein field equations: $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi G T_{\mu\nu}$
  • Sectional curvature
  • Geodesic deviation equation

Properties of Affine Connections

Metric Compatibility and Its Implications

• Metric compatibility ensures the inner product of vectors remains constant under parallel transport
• Defined by the condition $\nabla_X g(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$ for all vector fields X, Y, and Z
• Equivalent to $\nabla_X g = 0$ for all vector fields X
• In component notation: $\nabla_k g_{ij} = 0$
• Implications of metric compatibility:
  • Preserves lengths and angles during parallel transport
  • Simplifies calculations in many geometric contexts
  • Levi-Civita connection is the unique torsion-free, metric-compatible connection
• Relation to Christoffel symbols: $\Gamma^k_{ij} = \frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$

Symmetry of Connection and Its Consequences

• Symmetry of connection refers to the property $\Gamma^i_{jk} = \Gamma^i_{kj}$
• Equivalent to the vanishing of the torsion tensor: $T^i_{jk} = 0$
• Consequences of symmetric connections:
  • Parallel transport around infinitesimal closed loops is trivial to first order
  • Geodesics are reversible: if γ(t) is a geodesic, so is γ(-t)
  • Simplifies the form of the curvature tensor
• Levi-Civita connection is both symmetric and metric-compatible
• Non-symmetric connections arise in:
  • Theories with spin-orbit coupling
  • Some models of continuum mechanics
  • Certain approaches to quantum gravity
• Relation to the Lie bracket of vector fields: $[X,Y] = \nabla_X Y - \nabla_Y X$ for torsion-free connections