A connection answers a deceptively difficult question: how should a vector at one point be compared with a vector at another point when the two vectors live in different fibres?
On ordinary Euclidean space we compare vectors at different points by sliding them without changing their components. That rule is so familiar that it can disappear from view. On a general manifold or vector bundle, there is no automatic identification between fibres. A connection supplies a controlled notion of differentiation and parallel transport.
Curvature appears when this comparison depends on the route. Transport a vector around a small loop and it may return rotated. Differentiate a section in two directions and the order may matter. Curvature measures that failure of second derivatives to commute after the connection has been taken into account.
This guide follows Riemannian Geometry and Geodesics and the earlier Tangent, Cotangent and Tensor Fields guide. It treats connections first on a general vector bundle, then specialises to the tangent bundle and the Levi-Civita connection.
Level: undergraduate-to-beginning-graduate differential geometry. Prerequisites are smooth manifolds, vector bundles at an introductory level, differential forms, tensor fields and Riemannian metrics. This is enrichment rather than a school syllabus route. Use the BTT Mathematics Hub for level-specific study.
Reading route: why a connection is needed → covariant derivatives → local connection forms → parallel transport → curvature → tangent-bundle geometry → Bianchi identities → practice → solutions.
1. Different fibres cannot be compared by subtraction alone
Let E→M be a vector bundle. A section s assigns a vector s(p) in the fibre Eₚ over each point p. If q is near p, the expression s(q)−s(p) is not intrinsically defined because the two vectors belong to different vector spaces.
In a local trivialisation E|ᵁ≈U×Rᵐ, fibres are temporarily identified with the same model vector space and ordinary component derivatives become available. But another trivialisation changes those components by a point-dependent invertible matrix. Ordinary derivatives then acquire extra terms from differentiating that matrix.
A connection is precisely the extra structure that corrects those derivatives so the resulting object transforms consistently. It is not merely notation for partial derivatives. It records a rule for how neighbouring fibres are related.
A broad treatment separating bundles, connections and curvature is available in Chris Wendl’s Lecture Notes on Bundles and Connections. A graduate course sequence covering connections, parallel transport, curvature and Chern classes is outlined on the UC San Diego Math 250B page.
2. A connection differentiates sections in vector-field directions
A connection ∇ on E assigns to a vector field X and section s another section ∇ₓs. It is R-linear in s, C∞(M)-linear in X, and satisfies the Leibniz rule
∇ₓ(fs)=X(f)s+f∇ₓs.
C∞-linearity in X means (∇ₓs)(p) depends only on X(p), but it depends on the behaviour of s near p because differentiation of s is involved.
The connection may also be viewed as a first-order differential operator ∇:Γ(E)→Ω¹(M;E), where Ω¹(M;E) denotes E-valued 1-forms. Evaluating the resulting E-valued 1-form on X recovers ∇ₓs.
Connections always exist on smooth vector bundles over ordinary paracompact manifolds; partitions of unity allow local flat connections to be patched. Existence does not mean uniqueness. The space of connections is affine: the difference of two connections is an End(E)-valued 1-form.
3. The flat connection on a trivial bundle is the baseline
On the trivial bundle M×Rᵐ with global frame e₁,…,eₘ, write s=sᵃeₐ. The flat connection in this frame is ∇ₓs=X(sᵃ)eₐ. It differentiates only the component functions.
Parallel sections have constant components in this frame. Curvature is zero because ordinary mixed derivatives commute after the Lie-bracket correction is included.
But this rule is tied to the chosen global frame. If the frame itself is changed in a position-dependent way, the components of the same flat connection are no longer zero. Vanishing connection coefficients are therefore representation-dependent.
This resembles the earlier polar-coordinate example: Euclidean space is flat even though its Christoffel symbols in polar coordinates are not all zero.
4. A local frame packages a connection into a matrix of 1-forms
Choose a local frame e=(e₁,…,eₘ) and write the section component column as s. Define connection 1-forms Aᵃ_b by ∇e_b=Aᵃ_b⊗eₐ. In matrix notation,
∇s=ds+A s.
Here A is an m×m matrix whose entries are ordinary 1-forms. Evaluating on a vector field X gives ∇ₓs=X(s)+A(X)s.
If a new frame is related by e′=e g for a smooth invertible matrix g, the section components become s′=g⁻¹s. Requiring the same geometric connection gives
A′=g⁻¹Ag+g⁻¹dg.
The inhomogeneous g⁻¹dg term is why connection coefficients are not tensor components. It exactly compensates for the derivative of the changing frame.
5. Worked frame change: zero coefficients can become nonzero
Take the trivial line bundle over R with flat connection A=0 in frame e. Choose a new frame e′=e eˣ. Then g=eˣ, so g⁻¹dg=dx. The same flat connection has A′=dx.
A section s=ce in the original parallel frame has new component s′=ce⁻ˣ. Its covariant derivative is ds′+A′s′=−ce⁻ˣdx+ce⁻ˣdx=0.
If we differentiated only the new component, we would incorrectly conclude that the formerly parallel section is changing. The connection coefficient corrects the frame motion.
This one-dimensional example has no curvature, yet nonzero connection coefficients. It is the cleanest possible warning against equating “nonzero connection” with “curved geometry”.
6. Parallel transport solves an ODE along a curve
Let γ(t) be a smooth curve and let V(t) be a vector in the fibre E_{γ(t)}. It is parallel when ∇_{γ′}V=0.
In a local frame, this becomes the matrix ODE
V̇+A(γ′)V=0.
Given an initial vector V(a), standard ODE theory gives a unique parallel vector along the curve while the curve remains in the coordinate patch; overlapping trivialisations agree because the connection transformation law preserves the equation.
The resulting linear isomorphism P_γ:E_{γ(a)}→E_{γ(b)} is parallel transport. Reversing the path gives its inverse. Concatenating paths composes the corresponding maps.
Parallel transport depends on the connection. It may also depend on the path. Curvature measures the infinitesimal version of that dependence.
7. Metric connections preserve inner products during transport
If E has a fibre metric h, a connection is metric-compatible when X[h(s,t)]=h(∇ₓs,t)+h(s,∇ₓt). Along a curve, two parallel sections then have constant inner product.
Therefore parallel transport for a metric connection is an isometry between the endpoint fibres. Lengths and angles of transported vectors are preserved.
In an orthonormal frame for a real metric bundle, the connection matrix A is skew-symmetric: A+Aᵀ=0. For a Hermitian complex bundle, the analogous matrix is skew-Hermitian.
This converts an abstract compatibility condition into a concrete algebraic check on the local connection forms.
8. Curvature is the failure of covariant derivatives to commute
For a connection on E, define the curvature by
R(X,Y)s=∇ₓ∇ᵧs−∇ᵧ∇ₓs−∇_{[X,Y]}s.
This guide uses that sign convention. Some sources reverse the overall sign. The Lie-bracket term removes the failure of the vector fields themselves to commute, leaving a quantity that is C∞-linear in X, Y and s at the point.
Thus curvature is an End(E)-valued 2-form. It is tensorial even though the connection coefficients from which it is built are not.
If R=0, the connection is flat. Flatness is a local differential condition. It does not automatically mean parallel transport around every global loop is trivial when the base has nontrivial topology.
9. The local curvature formula is F=dA+A∧A
In a local frame with connection matrix A, the curvature matrix of 2-forms is
F=dA+A∧A.
The product combines matrix multiplication with wedge product: (A∧A)ᵃ_b=Σ_c Aᵃ_c∧Aᶜ_b. It need not vanish even though a scalar 1-form wedges with itself to zero, because different matrix entries are being multiplied and summed.
Under a frame change e′=eg, the curvature transforms homogeneously:
F′=g⁻¹Fg.
The troublesome derivative term has disappeared. This is exactly what tensorial geometric information should do.
The formula and its intrinsic meaning are standard in the connection notes listed by Wendl and in the MIT Geometry of Manifolds lecture on metric connections and curvature.
10. Worked curvature on a line bundle: the matrix term disappears
For a real or complex line bundle in a local frame, A is a scalar-valued 1-form. Since A∧A=0, the curvature is simply F=dA.
Take A=xdy on the trivial line bundle over R². Then F=dx∧dy. The connection is not flat.
Take instead A=df for any smooth scalar f. Then F=d²f=0. In fact the frame change g=e^{−f} removes this exact connection form locally: A′=A+g⁻¹dg=df−df=0.
On a domain with nontrivial topology, a closed connection 1-form need not be globally removable by a single-valued gauge function. Flatness controls local curvature; holonomy can still carry global information.
11. Holonomy records what transport does around loops
For a loop γ based at p, parallel transport gives a linear automorphism P_γ:Eₚ→Eₚ. The collection generated by such loop transports is the holonomy group of the connection at p.
For a flat connection on a simply connected region, parallel transport is path-independent and holonomy is trivial there. On a non-simply-connected base, flat connections can have nontrivial holonomy determined by loop classes.
For example, on a trivial complex line bundle over the circle with angular coordinate θ, a flat connection A=iλdθ has zero curvature. Parallel transport once around the circle multiplies a vector by exp(−2πiλ). Unless λ is an integer under this normalisation, the holonomy is nontrivial.
The example separates curvature from global monodromy. Zero local curvature does not erase topology.
12. A connection on the tangent bundle also has torsion
When E=TM, a connection compares tangent vectors at neighbouring points. In addition to curvature, it has a torsion tensor
T(X,Y)=∇ₓY−∇ᵧX−[X,Y].
Torsion measures the antisymmetric part of the connection relative to the Lie bracket. In a coordinate basis, [∂ᵢ,∂ⱼ]=0, so Tᵏ_{ij}=Γᵏ_{ij}−Γᵏ_{ji}.
The Levi-Civita connection of a Riemannian metric is the unique tangent-bundle connection that is both torsion-free and metric-compatible. The previous article used this uniqueness to define geodesics and metric curvature.
Other geometric settings deliberately use connections with torsion. Therefore “connection” should not silently be replaced by “Levi-Civita connection” unless a Riemannian metric and the two defining conditions have been specified.
13. Christoffel symbols are connection coefficients in a coordinate frame
For a tangent-bundle connection, write ∇_{∂ᵢ}∂ⱼ=Γᵏ_{ij}∂ₖ. The Γᵏ_{ij} are the Christoffel symbols of the connection in that coordinate frame.
For the Levi-Civita connection they are computed from the metric by Γᵏ_{ij}=(1/2)gᵏˡ(∂ᵢg_{jℓ}+∂ⱼg_{iℓ}−∂ℓg_{ij}). Their symmetry in i and j reflects zero torsion in a coordinate basis.
Under coordinate changes, the Γ symbols acquire an inhomogeneous second-derivative term. Consequently one can make them vanish at a chosen point using normal coordinates.
Curvature components do not transform that way. If curvature is nonzero, no coordinate change can make the curvature tensor itself vanish at that point.
14. The Riemann tensor measures the commutator of Levi-Civita derivatives
For the Levi-Civita connection, the same curvature definition gives the Riemann tensor R(X,Y)Z. In coordinates,
Rˡ_{kij}=∂ᵢΓˡ_{jk}−∂ⱼΓˡ_{ik}+Γˡ_{im}Γᵐ_{jk}−Γˡ_{jm}Γᵐ_{ik}
under the convention used here. Index placements and sign order differ across sources, so a formula should always travel with its defining convention.
Lowering the first index with the metric produces R_{ℓkij}. The Levi-Civita tensor has symmetries arising from metric compatibility and zero torsion, including antisymmetry in each index pair and pair interchange symmetry.
Those symmetries dramatically reduce the number of independent components. They are structural information, not cosmetic index identities.
15. Worked curvature check: Euclidean polar coordinates are flat
For the Euclidean plane g=dr²+r²dθ², nonzero Levi-Civita symbols include Γʳ_{θθ}=−r and Γ^θ_{rθ}=Γ^θ_{θr}=1/r.
Compute Rʳ_{θrθ}. Using the stated convention, the derivative term ∂ᵣΓʳ_{θθ}=−1 appears, while the relevant quadratic product contributes +1. The remaining terms vanish, giving zero.
This cancellation is the point. Connection coefficients describe how the chosen frame changes. Curvature tests whether those changes contain irreducible geometric path-dependence.
Since Euclidean space is globally flat, every component of the Riemann tensor vanishes in every coordinate system, even though the Christoffel symbols may be complicated.
16. Sphere curvature cannot be removed by coordinates
The round sphere of radius R has constant sectional curvature 1/R². In two dimensions the entire Riemann tensor is determined by this Gaussian curvature.
At any selected point we can use normal coordinates to set Γᵏ_{ij}=0 there. Yet derivatives of the connection coefficients remain arranged so that the Riemann tensor is nonzero.
Parallel transport gives an operational picture. Transport a tangent vector around a sufficiently small closed loop on a curved surface. The returned direction differs from the starting direction by an amount whose leading behaviour is controlled by curvature times oriented area.
For finite loops on surfaces, Gauss–Bonnet relates accumulated Gaussian curvature to boundary turning and topology. The small-loop transport picture is therefore the local beginning of a much broader global relationship.
17. Geodesic deviation is curvature acting on nearby trajectories
If γₛ is a family of geodesics and J is its variation field, then J satisfies the Jacobi equation ∇ₜ²J+R(J,γ′)γ′=0.
Thus curvature can be read as relative acceleration between neighbouring geodesics. On a sphere, suitable perpendicular geodesics reconverge; in negative-curvature models they tend to separate more rapidly.
The equation is not a slogan that every positive-curvature space forces every pair of paths to meet. It is a precise second-order equation whose conclusions depend on the sectional curvature in the relevant planes and on the initial conditions.
This is one bridge from local curvature tensors to global comparison geometry.
18. Ricci curvature and scalar curvature are controlled contractions
The Ricci tensor contracts the Riemann tensor, tracing how curvature acts across an orthonormal set of transverse directions. Scalar curvature then traces Ricci with the metric.
These contractions are invariant because the metric supplies the legitimate upper-lower index pairing. Simply summing a convenient set of curvature-matrix entries without the correct tensor contraction would be coordinate-dependent.
Ricci curvature controls important averaged geometric behaviour such as volume comparison and geodesic focusing. Scalar curvature is an even coarser contraction. Neither contains all the information of sectional curvature in dimensions four and above.
Different geometric problems ask for different levels of curvature information. The calculation should name which one is relevant before compressing the tensor.
19. Bianchi identities constrain curvature
For a general connection, the curvature satisfies a covariant exterior identity often written d_∇F=0. In a local matrix frame this becomes dF+A∧F−F∧A=0.
For the Levi-Civita connection there is also the algebraic first Bianchi identity R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0.
Contracting the differential Bianchi identity produces the divergence-free Einstein tensor in Riemannian or pseudo-Riemannian geometry. That application belongs to geometric physics, but the underlying identity is purely differential-geometric.
The Bianchi identities show that curvature components cannot be chosen independently. They are linked by the fact that curvature itself was built from a connection.
20. Curvature forms lead toward characteristic classes
For a complex vector bundle with connection, invariant polynomials in the curvature matrix produce closed differential forms. Chern–Weil theory shows that their de Rham cohomology classes do not depend on the chosen connection.
This is a remarkable local-to-global bridge. The connection is flexible auxiliary data, while the resulting characteristic cohomology class can encode topological information about the bundle.
The next Atlas cell develops this more carefully through vector bundles, sections, characteristic classes and global analysis. The UCSD graduate course outline makes the same progression from connections and curvature to Chern classes.
This transition should not be compressed into “curvature equals topology”. The curvature form depends on the chosen connection; specific invariant combinations represent connection-independent cohomology classes.
21. A dependable connection calculation workflow
First identify the bundle, base manifold, local frame and tensor type. State whether the connection is arbitrary, metric-compatible, Levi-Civita or another specialised connection.
For a frame change, transform section components and the connection matrix together. Verify a scalar or fibre-metric quantity after the transformation.
For curvature, calculate F=dA+A∧A rather than differentiating entries without matrix order. On the tangent bundle, record the sign convention for R and the index convention before comparing formulas.
Finally, distinguish local flatness from global trivial holonomy, connection coefficients from curvature, and an arbitrary connection from the Levi-Civita connection. These three separations catch a large fraction of conceptual errors.
22. Independent practice: fourteen questions
- On a trivial line bundle over R, start with A=0 and change frame by g=e²ˣ. Find A′.
- For the result of Question 1, show that the section with new component e⁻²ˣ is parallel.
- For A=xdy on a line bundle over R², compute F.
- For A=df on a line bundle, prove F=0.
- For the matrix connection A=[[0,−xdy],[xdy,0]], compute dA and A∧A.
- Explain why F transforms tensorially while A does not.
- For A=iλdθ on a complex line bundle over S¹, compute one-turn holonomy under the convention V̇+A(γ′)V=0.
- Why can a flat connection have nontrivial holonomy on S¹?
- In a coordinate frame, express torsion components in terms of Γᵏ_{ij}.
- Why are the lower two Christoffel indices symmetric for the Levi-Civita connection in coordinates?
- Using Euclidean polar Christoffel symbols, compute Rʳ_{θrθ} and verify zero.
- State one reason curvature can remain nonzero in normal coordinates at a point.
- For a round sphere of radius R, state sectional, Ricci and scalar curvature.
- Explain why changing the connection can change curvature forms while a Chern–Weil cohomology class can remain unchanged.
23. Worked solutions and checks
1. With A=0 and scalar g=e²ˣ, A′=g⁻¹dg=2dx.
2. If s′=e⁻²ˣ, then ds′=−2e⁻²ˣdx. Adding A′s′ gives −2e⁻²ˣdx+2e⁻²ˣdx=0.
3. On a line bundle A∧A=0, so F=d(xdy)=dx∧dy.
4. Again A∧A=0 and dA=d²f=0.
5. Differentiating gives dA=[[0,−dx∧dy],[dx∧dy,0]]. Every entry of A is proportional to the same 1-form xdy, so the matrix product A∧A has factors dy∧dy and is zero. Hence F=dA.
6. A′ contains the extra derivative term g⁻¹dg because the frame itself varies. In F=dA+A∧A, those inhomogeneous terms cancel, leaving F′=g⁻¹Fg.
7. Along θ from 0 to 2π, the parallel equation is V̇+iλV=0. Thus V(2π)=e⁻²πiλV(0).
8. Curvature is local and vanishes, but loops around S¹ are not contractible through loops in S¹. Parallel transport can therefore encode a nontrivial representation of the fundamental group.
9. Because coordinate basis fields commute, Tᵏ_{ij}=Γᵏ_{ij}−Γᵏ_{ji}.
10. Levi-Civita torsion is zero, so Tᵏ_{ij}=0 and Γᵏ_{ij}=Γᵏ_{ji} in a coordinate frame.
11. Using Γʳ_{θθ}=−r and Γ^θ_{rθ}=1/r, the derivative contribution is −1 and the relevant quadratic term contributes +1, with other terms zero. Their sum is zero, as required by Euclidean flatness.
12. Normal coordinates can set the connection coefficients themselves to zero at p, but curvature also contains first derivatives of those coefficients. Those derivatives encode second-order metric information that cannot generally be removed.
13. Sectional curvature is 1/R². In dimension two, Ric=(1/R²)g and scalar curvature is 2/R² under the stated sign convention.
14. Different connections have different curvature matrices. Chern–Weil theory applies invariant polynomials and proves that the difference between the resulting closed forms for two connections is exact, so the de Rham cohomology class is unchanged.
24. Questions that separate local, global and representational effects
Does A=0 in one frame mean the connection is globally trivial?
Only on the region covered by that frame. A different frame may produce nonzero coefficients, and global topology may prevent one frame from existing everywhere.
Does F=0 mean all loop transport is identity?
Locally on simply connected regions, flatness gives path independence. Globally, a flat connection can have nontrivial holonomy around noncontractible loops.
Does curvature require a Riemannian metric?
No. Any connection on a vector bundle has curvature. A Riemannian metric supplies the Levi-Civita connection and lets the tangent-bundle curvature be contracted into sectional, Ricci and scalar curvature.
What comes next?
Connections live on bundles, and their curvature can create characteristic forms. The next guide develops vector bundles, sections, clutching, characteristic classes and the beginning of global analysis.
Sources and further study
Chris Wendl’s Lecture Notes on Bundles and Connections provide an extended route through bundles, connections and curvature. The UC San Diego Math 250B course page lays out connections, parallel transport, curvature, holonomy and Chern classes. MIT OpenCourseWare Geometry of Manifolds, Lecture 9 covers metric connections and intrinsic curvature of a connection.
Continue to bundles, characteristic structures and global analysis. Return to the BTT Mathematics Hub.