Mathematics > Differential Geometry > Affine Connections

Affine Connections in Differential Geometry

Differential geometry is a vibrant field of mathematics concerned with the study of smooth shapes and the properties of spaces that are curved rather than flat. Within this domain, affine connections stand out as a critical concept, providing a rigorous way to compare vectors at different points on a manifold—a type of space that locally resembles Euclidean space.

An affine connection, or simply a connection, on a smooth manifold \( M \) allows for the differentiation of vector fields along the manifold. It essentially provides a way to transport vectors smoothly from one point to another in a manner that respects the curvature and structure of the manifold. This process is known as parallel transport.

Mathematically, an affine connection is often denoted by \( \nabla \), called the covariant derivative. It is a bilinear operator that acts on pairs of smooth vector fields \( X \) and \( Y \) on \( M \), and maps them to another smooth vector field \( \nabla_X Y \). The covariant derivative \(\nabla_X Y\) intuitively represents the rate of change of \( Y \) in the direction of \( X \).

One of the main properties of a connection is how it respects the manifold’s smooth structure:
1. Linearity: For any smooth functions \( f \) and \( g \) on the manifold \( M \), and any vector fields \( X, Y, Z \),
\[
\nabla_{fX + gY} Z = f \nabla_X Z + g \nabla_Y Z.
\]
2. Leibniz Rule: For any smooth function \( f \) and any pair of vector fields \( X \) and \( Y \),
\[
\nabla_X (fY) = X(f)Y + f\nabla_X Y,
\]
where \( X(f) \) denotes the action of the vector field \( X \) on the function \( f \).

Affine connections allow the definition of several important geometric concepts such as curvature and torsion. Curvature gives a measure of how the manifold bends, and is encapsulated by the Riemann curvature tensor \( R \), defined by
\[
R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z,
\]
where \( [X, Y] \) is the Lie bracket of the vector fields \( X \) and \( Y \).

Torsion measures the failure of the connection to be symmetric and is described by the torsion tensor \( T \), defined as
\[
T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y].
\]

Affine connections are fundamental in various branches of geometry and physics, particularly in the theory of general relativity, where the connection used is the Levi-Civita connection, defined uniquely by being torsion-free and metric-compatible (it preserves the inner product of vectors).

Understanding affine connections enables deeper insights into the behavior and properties of curved spaces and forms the foundation for more advanced studies in Riemannian geometry, complex manifolds, and the geometric theory of differential equations.