Connections and Curvature

A connection is a rule for differentiating and comparing data that live in different fibers over a manifold. Curvature measures the failure of that comparison rule to be path independent.

The slogan is:

A connection tells you how to move nearby; curvature tells you whether small loops come back unchanged.

This page gives the local formulas needed for Berry connections, gauge potentials, tangent-vector transport, and later topology pages. It uses the word “fiber” informally: a fiber is a vector space or phase line attached to each point of a base manifold. The full bundle language is introduced in Fiber Bundles, First Look.

Helpful background. Review Tangent and Cotangent Spaces for local bases and Exterior Derivative for differentiating forms.

Comparing fibers at nearby points

Suppose each value of a control parameter has its own eigenvector. Even after normalizing it, its phase is arbitrary, so subtracting neighboring representatives need not measure a physical change. A connection corrects for changes of local frame and makes comparison along a path meaningful.

From Ordinary to Covariant Derivatives

For a scalar function $f$ on a manifold, the ordinary differential $df$ is intrinsic. For a vector-like object $s$ whose components live in a fiber, differentiating the component functions is not enough, because a change of local frame changes the meaning of those components.

Let $e_a$ be a local frame for the fibers, and write a section as

$$s=s^a e_a.$$

A connection specifies how the frame itself changes:

$$\nabla_i e_a = \Gamma^b{}_{ia}e_b.$$

Here $\nabla_i$ means covariant differentiation in the coordinate direction $\partial_i$, and the coefficients $\Gamma^b{}_{ia}$ are the local connection coefficients.

The covariant derivative of $s$ is then

$$\nabla_i s = \left( \partial_i s^b+\Gamma^b{}_{ia}s^a \right)e_b.$$

The first term differentiates the component functions. The second term corrects for the change of frame.

Connection One-Forms

It is often cleaner to package the coefficients into a matrix-valued one-form:

$$\Gamma^b{}_a = \Gamma^b{}_{ia}\,dx^i.$$

In matrix notation,

$$\nabla s = ds+\Gamma s.$$

This compact expression should be read in a chosen local frame. Under a frame change, $\Gamma$ changes by a transformation law that includes an inhomogeneous derivative term. For that reason, connection coefficients are not tensor components.

The covariant derivative $\nabla s$, however, is geometric once the connection has been chosen.

Tangent-Bundle Example

For the tangent bundle of a manifold, a vector field is written

$$V=V^i\partial_i.$$

A connection on the tangent bundle is locally described by coefficients $\Gamma^k{}_{ij}$:

$$\nabla_i\partial_j = \Gamma^k{}_{ij}\partial_k.$$

Therefore

$$\nabla_i V^k = \partial_i V^k+\Gamma^k{}_{ij}V^j.$$

If the manifold has a Riemannian metric, the most common connection is the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric. But a connection is not automatic from the smooth manifold alone. It is extra structure, unless a metric or gauge principle specifies it.

Parallel Transport Preview

Let $\gamma(t)$ be a path with local coordinates $x^i(t)$. A section $s(t)$ along the path is parallel transported if

$$\nabla_{\dot\gamma}s=0.$$

In components this is the ordinary differential equation

$$\frac{ds^b}{dt} + \dot x^i(t)\Gamma^b{}_{ia}(x(t))s^a(t) = 0.$$

Thus a connection turns a path into a transport rule. Parallel Transport focuses on the path-ordered solution and the geometric-phase intuition. For this page, the important point is that parallel transport is not defined before a connection is specified.

Curvature as Noncommuting Covariant Derivatives

Curvature measures the failure of covariant derivatives to commute. In a vector bundle, the curvature acts on a section by

$$[\nabla_i,\nabla_j]s = F_{ij}s,$$

where $F_{ij}$ is matrix-valued in the fiber indices.

Using connection one-forms, the curvature two-form is

$$F = d\Gamma+\Gamma\wedge\Gamma.$$

In components,

$$F^b{}_a = \frac12 F^b{}_{aij}\,dx^i\wedge dx^j,$$

with

$$F^b{}_{aij} = \partial_i\Gamma^b{}_{ja} - \partial_j\Gamma^b{}_{ia} + \Gamma^b{}_{ic}\Gamma^c{}_{ja} - \Gamma^b{}_{jc}\Gamma^c{}_{ia}.$$

The derivative terms say that the local connection changes from point to point. The quadratic terms say that different fiber rotations may fail to commute.

Sign and index conventions vary across geometry, gauge theory, and relativity. This page uses the convention in which $F=d\Gamma+\Gamma\wedge\Gamma$ and $[\nabla_i,\nabla_j]s=F_{ij}s$.

Abelian Case

For a Hermitian complex line bundle with a unitary connection, write $\Gamma=-iA$, where $A$ is a real one-form in a normalized frame. The matrix-valued curvature used above is then $\mathcal F=d\Gamma=-i\,dA$. It is customary to call the real two-form $F=i\mathcal F$ the phase curvature, so

$$F=dA.$$

This is the case behind the simplest Berry connection. It is also the differential-form pattern behind electromagnetism, where a gauge potential one-form has a field-strength two-form.

Because $d^2=0$, a gauge shift of the form

$$A\mapsto A-d\chi$$

leaves the curvature invariant:

$$d(A-d\chi) = dA.$$

The connection is gauge dependent. The curvature is gauge invariant in the abelian case.

Berry Connection Preview

Suppose a Hamiltonian $H(R)$ has a smooth nondegenerate normalized eigenstate

$$H(R)\lvert n(R)\rangle = E_n(R)\lvert n(R)\rangle$$

on a patch of parameter space. The eigenstate is not unique: it may be changed by a parameter-dependent phase,

$$\lvert n(R)\rangle \mapsto e^{i\chi(R)}\lvert n(R)\rangle.$$

The local Berry connection one-form is

$$A_n = i\langle n(R)\rvert d n(R)\rangle.$$

In coordinates,

$$A_n = A_i^{(n)}(R)\,dR^i, \qquad A_i^{(n)} = i\langle n(R)\rvert\partial_i n(R)\rangle.$$

Under the phase change above,

$$A_n\mapsto A_n-d\chi.$$

The Berry curvature is therefore

$$F_n=dA_n.$$

In coordinates,

$$F_n = \frac12 F_{ij}^{(n)}\,dR^i\wedge dR^j,$$

where

$$F_{ij}^{(n)} = \partial_iA_j^{(n)} - \partial_jA_i^{(n)}.$$

The physical adiabatic phase around a loop is discussed in Berry Phase. Berry Connection as a Mathematical Object derives the gauge transformation and a curvature formula that uses the eigenprojector directly.

Curvature and Small Loops

Curvature can be detected by transporting around an infinitesimal loop. In a local coordinate patch, take a tiny rectangle spanned by coordinate displacements $\delta x^i$ and $\delta x^j$. Parallel transport around the rectangle returns a section approximately as

$$s \mapsto s - \mathcal F_{ij}s\,\delta x^i\delta x^j$$

for the traversal $+i,+j,-i,-j$, to leading order in the area, where $\mathcal F=d\Gamma+\Gamma\wedge\Gamma$. In the unitary line case this is $s\mapsto(1+iF_{ij}\,\delta x^i\delta x^j)s$, consistent with the phase $e^{i\int F}$. Reversing the loop reverses the first-order sign.

The operational meaning is simple: if curvature vanishes on a simply connected patch, parallel transport is locally path independent after a suitable choice of frame. If curvature is nonzero, two nearby routes between the same endpoints generally give different fiber data.

Global topology can still matter when curvature vanishes locally. A flat connection on a space with noncontractible loops can have nontrivial holonomy, and winding is the simplest way to label such loop classes. Topological invariants are one reason topology and gauge structure cannot be replaced by local curvature alone.

Worked Example: Constant Abelian Curvature

On the $xy$-plane, define the one-form

$$A = \frac{B}{2} \left( -y\,dx+x\,dy \right).$$

Compute its curvature:

$$\begin{aligned} F &= dA\\ &= \frac{B}{2} \left( -dy\wedge dx+dx\wedge dy \right)\\ &= B\,dx\wedge dy. \end{aligned}$$

The connection one-form $A$ depends on a gauge choice. The curvature two-form $F$ is the invariant local flux density.

For a rectangle $\Sigma=[0,L_x]\times[0,L_y]$ with the standard orientation,

$$\int_\Sigma F = BL_xL_y.$$

If $A$ is smooth on the rectangle, Stokes theorem gives

$$\oint_{\partial\Sigma}A = \int_\Sigma F = BL_xL_y.$$

This is the prototype for “phase equals curvature flux” formulas.

Sphere Example: Curvature Without a Global Frame

On a sphere, a tangent vector can be parallel transported along a curve using the Levi-Civita connection. Around a closed loop, the vector may return rotated relative to its initial direction. The rotation is determined by the curvature enclosed by the loop, with orientation and sign conventions fixed by the transport rule.

This example is useful for Berry phase intuition because it shows that local rules can produce global effects around loops. But the analogy should be handled carefully: the Levi-Civita connection transports tangent vectors on a curved surface, while the Berry connection transports phase choices in a complex line over parameter space.

Common Mistakes

• Comparing vectors, phases, or eigenspaces at different base points without specifying a connection.
• Treating connection coefficients $\Gamma^b{}_{ia}$ as tensor components.
• Assuming a smooth manifold automatically comes with a preferred connection.
• Confusing a connection one-form with its curvature two-form.
• Thinking gauge dependence of $A$ makes curvature or closed-loop phase meaningless.
• Assuming zero local curvature always implies trivial global holonomy.
• Applying the sphere tangent-vector analogy to Berry phase without noting that the fibers are different.
• Forgetting that nonabelian curvature includes the $\Gamma\wedge\Gamma$ term.

Exercises

1. In a local frame with $\Gamma=0$, solve the parallel-transport equation along a curve.

Solution

The transport equation is

$$\frac{ds^b}{dt} + \dot x^i\Gamma^b{}_{ia}s^a = 0.$$

If $\Gamma=0$, this becomes

$$\frac{ds^b}{dt}=0.$$

Thus every component $s^b$ is constant along the curve. In this frame, parallel transport keeps the component vector fixed.

2. In one dimension, let a line-bundle connection be $\Gamma(t)\,dt$. Solve

$$\frac{ds}{dt}+\Gamma(t)s=0.$$

Solution

For $s(t)\ne0$,

$$\frac{1}{s}\frac{ds}{dt} = -\Gamma(t).$$

Integrating from $0$ to $t$ gives

$$\log\frac{s(t)}{s(0)} = - \int_0^t\Gamma(\tau)\,d\tau.$$

Therefore

$$s(t) = s(0) \exp \left( - \int_0^t\Gamma(\tau)\,d\tau \right).$$

3. Compute the curvature of

$$A = \frac{B}{2} \left( -y\,dx+x\,dy \right).$$

Solution

Use $d(dx)=d(dy)=0$ and $d^2=0$:

$$\begin{aligned} dA &= \frac{B}{2} \left( -dy\wedge dx+dx\wedge dy \right)\\ &= \frac{B}{2} \left( dx\wedge dy+dx\wedge dy \right)\\ &= B\,dx\wedge dy. \end{aligned}$$

4. If $A\mapsto A-d\chi$ in the abelian case, show that $F=dA$ is invariant.

Solution

The transformed curvature is

$$F' = d(A-d\chi) = dA-d^2\chi.$$

Since $d^2=0$,

$$F'=dA=F.$$

5. Explain why the connection coefficients can vanish at one point in a suitable frame while curvature may still be nonzero there.

Solution

Connection coefficients are not tensor components, so a frame can often be chosen to make them vanish at a chosen point. Curvature is tensorial. It depends on the first-order failure of the connection to be removable around infinitesimal loops, not merely on the value of the connection coefficients at one point. Thus $\Gamma(p)=0$ in a frame does not imply $F(p)=0$.

6. Why does Berry phase need more than the instantaneous energy eigenvalues?

Solution

The dynamical phase depends on the energies through

$$-\frac{1}{\hbar}\int E_n(t)\,dt.$$

The Berry phase depends on how the eigenvectors vary over parameter space, including their phase convention and connection. Two Hamiltonian paths can have the same instantaneous eigenvalues but different eigenvector geometry, and hence different Berry holonomy.

References

• T. Frankel, The Geometry of Physics, 3rd ed., Cambridge University Press, 2011.
• J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013.
• M. Nakahara, Geometry, Topology and Physics, 2nd ed., CRC Press, 2003.