Connections and Covariant Derivatives — Mathematics Knowledge System

Overview

A connection (or covariant derivative) on a smooth manifold is a rule for differentiating vector fields (and more generally tensor fields) along curves or in given directions, in a way that is coordinate-independent. On a Riemannian manifold there is a canonical connection -- the Levi-Civita connection -- that is compatible with the metric and torsion-free. Connections are the fundamental tool for parallel transport, defining geodesics, and computing curvature.

Intuition

On flat ℝⁿ, differentiating a vector field is trivial: just differentiate each component. On a curved manifold, this fails because tangent spaces at different points are different vector spaces -- you cannot directly compare vectors at different points. A connection provides a way to 'connect' nearby tangent spaces, enabling differentiation. Parallel transport along a curve uses the connection to move a vector along the curve while keeping it 'as constant as possible'. The failure of parallel transport to return a vector to its original value after a loop is measured by curvature.

Formal Definition

Definition

A connection (linear connection) on M is a map ∇: Γ(TM) × Γ(TM) → Γ(TM), written (X, Y) ↦ ∇_X Y, satisfying four axioms. The Levi-Civita connection is the unique connection that is (1) compatible with g (metric connection) and (2) torsion-free (symmetric).

\nabla_{fX} Y = f\, \nabla_X Y

Tensoriality in X (C∞(M)-linearity)

\nabla_X (fY) = (Xf)\, Y + f\, \nabla_X Y

Leibniz rule in Y

\nabla_X Y - \nabla_Y X = [X, Y]

Torsion-free condition (zero torsion)

X\, g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)

Metric compatibility

\nabla_{\partial_i} \partial_j = \Gamma^k_{ij}\, \partial_k

Connection coefficients (Christoffel symbols) in local coordinates

\Gamma^k_{ij} = \frac{1}{2} g^{kl}\left(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}\right)

Christoffel symbols for the Levi-Civita connection

Properties

Parallel transport is a linear isometry

\text{The parallel transport map } P_{\gamma}: T_{\gamma(a)}M \to T_{\gamma(b)}M \text{ is a linear isometry (preserves the metric).}

Covariant derivative of a tensor

(\nabla_X T)(Y_1,\ldots,Y_k) = X(T(Y_1,\ldots,Y_k)) - \sum_{i=1}^k T(Y_1,\ldots, \nabla_X Y_i, \ldots, Y_k)

Theorems

Theorem 1: Fundamental Theorem of Riemannian Geometry

\text{On every Riemannian manifold } (M,g), \text{ there exists a unique torsion-free metric-compatible connection} -- \text{the Levi-Civita connection.}

Theorem 2: Parallel transport along curves

\text{Given a curve } \gamma: [a,b] \to M \text{ and } v_0 \in T_{\gamma(a)}M, \text{ there exists a unique parallel vector field } V \text{ along } \gamma \text{ with } V(a) = v_0.

Worked Examples

1
From the Christoffel symbols of S² computed earlier:
$\Gamma^\theta_{\theta\phi} = 0,\quad \Gamma^\phi_{\theta\phi} = \cot\theta$
2
Apply the formula ∇_{∂_i} ∂_j = Γᵏᵢⱼ ∂_k:
$\nabla_{\partial/\partial\theta}\frac{\partial}{\partial\phi} = \Gamma^\theta_{\theta\phi}\frac{\partial}{\partial\theta} + \Gamma^\phi_{\theta\phi}\frac{\partial}{\partial\phi} = \cot\theta\frac{\partial}{\partial\phi}$

✓ Answer

∇_{∂θ}(∂φ) = cot θ · ∂φ. This reflects the fact that moving northward on the sphere, the longitudinal vector field 'fans out'.

Practice Problems

Hardproof writing

Prove that any connection ∇ on M defines a notion of parallel transport along any smooth curve.

Mediumfree response

What is the torsion tensor of a connection ∇, and why is the Levi-Civita connection torsion-free?

Common Mistakes

Common Mistake

The covariant derivative is the same as the directional derivative on ℝⁿ

On ℝⁿ with Cartesian coordinates the Christoffel symbols vanish, so ∇_X Y reduces to the directional derivative. On a curved manifold, the Christoffel symbol terms correct for the curvature of the coordinate system.

Common Mistake

The Christoffel symbols Γᵏᵢⱼ are components of a tensor

They are not -- they transform inhomogeneously under coordinate changes. Only the full covariant derivative ∇_X Y, or the curvature tensor built from second derivatives and products of Christoffel symbols, gives a genuine tensor.

Quiz

Historical Background

The covariant derivative was introduced by Ricci and Levi-Civita in their 1900 tensor calculus paper as a way to differentiate tensors on curved surfaces and manifolds. Levi-Civita introduced the concept of parallel transport in 1917, interpreting covariant differentiation geometrically. Hermann Weyl generalised connections in 1918 to study gauge theories. Élie Cartan reformulated connections in terms of differential forms and moving frames, laying the groundwork for modern principal bundle connections. Ehresmann's 1950 theory of connections on fibre bundles gave the most general framework.

1900
Ricci and Levi-Civita introduce the covariant derivative in tensor calculus
Gregorio Ricci-Curbastro, Tullio Levi-Civita
1917
Levi-Civita defines parallel transport and its relation to covariant differentiation
Tullio Levi-Civita
1918
Weyl introduces gauge connection in his unified field theory
Hermann Weyl
1950
Ehresmann defines connections on principal fibre bundles in full generality
Charles Ehresmann

Summary

A connection ∇ on M is a bilinear map (X, Y) ↦ ∇_X Y satisfying C∞-linearity in X and the Leibniz rule in Y.
The Levi-Civita connection is the unique torsion-free, metric-compatible connection on a Riemannian manifold.
In local coordinates, the connection coefficients are the Christoffel symbols Γᵏᵢⱼ determined by first derivatives of g.
Parallel transport uses the connection to move vectors along curves; holonomy measures the failure of loops to return vectors unchanged.
Curvature measures the failure of second covariant derivatives to commute: R(X,Y)Z = ∇_X∇_Y Z − ∇_Y∇_X Z − ∇_{[X,Y]} Z.

References

BookKobayashi, S. and Nomizu, K. -- Foundations of Differential Geometry, Vol. 1, Wiley, 1963
Bookdo Carmo, M. P. -- Riemannian Geometry, Birkhäuser, 1992, Chapter 2
WebsiteWikipedia -- Levi-Civita connection
WebsitenLab -- connection on a bundle