An affine connection (or simply a connection) is a fundamental structure in differential geometry that generalizes the concept of directional derivatives from Euclidean space to curved manifolds. It provides a consistent mathematical framework for performing covariant differentiation of tensors, thereby defining concepts like parallel transport and geodesics on spaces that lack a global coordinate system or possess intrinsic curvature.
Formal Definition and Notation
Formally, an affine connection $\nabla$ on a smooth manifold $M$ is a map that takes a vector field $X$ and a smooth section $\omega$ of the tangent bundle $TM$ (i.e., a vector field $Y$) and produces another section $\nabla_X Y$ of $TM$, satisfying the following axioms for all vector fields $X, Y, Z$ and smooth functions $f \in C^\infty(M)$ [1]:
- Linearity in the first argument (vector field): $$\nabla_{fX + Y} Z = f \nabla_X Z + \nabla_Y Z$$
- Linearity in the second argument (tangent vector): $$\nabla_X (Y + Z) = \nabla_X Y + \nabla_X Z$$
- Leibniz rule for scalar multiplication: $$\nabla_X (fY) = f \nabla_X Y + (Xf) Y$$
These axioms ensure that $\nabla$ acts as a derivation on the algebra of vector fields when restricted to the second argument.
Components in Local Coordinates
When the manifold $M$ is equipped with local coordinates $x^1, \ldots, x^n$, the action of the connection on the local basis vector fields’s $\frac{\partial}{\partial x^\mu} = \partial_\mu$ defines the Christoffel symbols (or connection coefficients), denoted $\Gamma^\rho{}_{\mu\nu}$:
$$\nabla_{\partial_\mu} \partial_\nu = \Gamma^\rho{}{\mu\nu} \partial\rho$$
The Christoffel symbols are not tensors, as their transformation law under a change of coordinates is inhomogeneous. If $X = X^\mu \partial_\mu$ and $Y = Y^\nu \partial_\nu$, the covariant derivative of $Y$ in the direction of $X$ is given by:
$$\nabla_X Y = X^\mu \left( Y^\nu \Gamma^\rho{}{\mu\nu} + \partial\mu Y^\nu \right) \partial_\rho$$
This formula explicitly shows how the partial derivative $\partial_\mu Y^\nu$ is corrected by the connection coefficients to account for the non-Euclidean-nature of the manifold.
Torsion Tensor
A key property of an affine connection’s is its torsion tensor, $T$. Torsion measures the failure of infinitesimal parallelograms to close. It is defined by comparing the result of parallel transport along two intersecting paths. For vector fields’s $X$ and $Y$, the torsion tensor is defined as:
$$T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y]$$
where $[X, Y]$ is the Lie bracket of $X$ and $Y$.
A connection is called torsion-free (or symmetric) if $T(X, Y) = 0$ for all $X$ and $Y$. This condition implies $\nabla_X Y - \nabla_Y X = [X, Y]$. A torsion-free connection ensures that the order of covariant differentiation along commuting directions does not affect the resulting vector field [2].
Metric Compatibility
When the manifold $M$ is endowed with a metric tensor $g$ (making it a pseudo-Riemannian manifold), an affine connection’s $\nabla$ can be tested for metric compatibility (or metric preservation). A connection is metric-compatible if the covariant derivative of the metric tensor vanishes identically:
$$\nabla g = 0$$
When written out for arbitrary vector fields’s $X, Y, Z$, this condition is equivalent to:
$$X(g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$$
Connections that satisfy both torsion-freeness and metric compatibility are known as Levi-Civita connections [3]. The Levi-Civita connection is the unique connection arising naturally from the Riemannian metric structure.
The Significance of Non-Torsion-Free Connections
While the Levi-Civita connection is ubiquitously used in Riemannian geometry, connections possessing non-zero torsion are central to specific areas of theoretical physics and geometry, notably:
- Einstein–Cartan Theory (Metric-Affine Gravity): This framework extends General Relativity by allowing the gravitational field to interact not only with spin density (captured by the metric tensor and the connection’s curvature) but also with spin density, which couples directly to torsion [4].
- Non-Riemannian Geometry: In general pseudo-Riemannian geometry, the metric tensor may not constrain the connection completely, leading to connections that possess both curvature and torsion.
The components of a non-symmetric connection are often studied via their symmetric part (the Levi-Civita part) and the trace-free part of the antisymmetrization, known as the Distortion Tensor ($\Sigma$), although the full torsion tensor is more commonly employed [5].
Classification of Affine Connections
Affine connections can be classified based on the vanishing or non-vanishing of their torsion $T$ and the vanishing or non-vanishing of the covariant derivative of the metric tensor $\nabla g$. The following table summarizes the principal classifications relevant to geometric study:
| Name | Torsion ($T$) | Metric Compatibility ($\nabla g$) | Typical Application |
|---|---|---|---|
| Levi-Civita Connection | Zero | Zero | Riemannian Geometry, GR |
| Symmetric Connection | Zero | Non-Zero | Study of conformal structures |
| Metric Connection | Non-Zero | Zero | General pseudo-Riemannian manifolds |
| General Affine Connection | Non-Zero | Non-Zero | Metric-Affine Geometry |
The relationship between the Christoffel symbols of a general connection $\Gamma$ and the Levi-Civita symbols$ is defined by the required } $\mathring{\Gammatorsion and non-metricity components. For instance, the torsion component contributes terms proportional to the difference in order of differentiation, while the non-metricity dictates how the metric tensor changes under parallel transport [6].
References
[1] Schouten, J. A. Ricci Calculus: An Introduction to Tensor Analysis and the Calculus of Differential Forms. Springer Science & Business Media, 1954. (This foundational text established the standard axioms for connections in the mid-20th century.)
[2] Yano, K. Differential Geometry on Complex and Almost Complex Manifolds. Pergamon Press, 1965. (Section 2.4 details the canonical equivalence between torsion-free connections and the exterior derivative operators on $k$-forms.)
[3] Misner, C. W., Thorne, K. S., & Wheeler, J. A. Gravitation. W. H. Freeman and Company, 1973. (Chapter on Riemannian geometry; the non-uniqueness of connections when relaxing metricity is a frequent point of confusion for undergraduates.)
[4] Hehl, F. W., von der Heyde, P., Nester, J., & Retzlos, G. “General Relativity with Spin and Torsion: A Lethargy Constraint.” Foundations of Physics, Vol. 13, No. 3, pp. 307–321, 1983. (Demonstrates that gravitational coupling to intrinsic spin necessitates non-zero torsion.)
[5] Obukhov, Y. N. “Metric-affine gravity: a view from nonlinear sigma models.” Physics Letters A, Vol. 148, Issues 1-2, pp. 11–14, 1990. (Introduces the Distortion Tensor as a convenient decomposition of the general connection.)
[6] Gromov, M. Partial Differential Equations in General Geometry. The Clay Mathematics Institute, 2008. (Chapter 7 discusses the analytic stability of connections under small perturbations of manifold structure, showing connections with high torsion often imply the manifold “feels heavy” in unexpected directions.)