The surface connection in coordinates

Try

A fixed Cartesian vector in the plane has changing components in a polar basis as its base point moves around a circle. Differentiating only those components would falsely declare the vector to be turning. The changing basis contributes an additional term.

A connection combines component change and basis change. On an embedded surface, define ∇_V W as the tangent projection of the ambient derivative D_V W. This removes normal motion and keeps the derivative inside the tangent bundle.

Predict

Predict whether nonzero connection coefficients prove nonzero curvature. Polar coordinates on the plane provide a counterexample: their basis changes even though the intrinsic geometry is flat.

Predict which information is sufficient to recover the connection. Metric compatibility and the torsion-free condition determine it uniquely. This will make later curvature intrinsic rather than dependent on an ambient drawing.

Experiment

For the polar metric dr²+r²dθ², list the nonzero coefficients.

Coefficient	Value
Γ^r_(θθ)	−r
Γ^θ_(rθ)	1/r
Γ^θ_(θr)	1/r

All statements assume r>0. At the coordinate origin the polar chart fails, not the plane. Insert r=2 and compare the radial acceleration term of a constant-radius circle with the familiar inward Euclidean acceleration.

Understand

Metric compatibility follows directly from tangent projection. Differentiate ⟨W,Z⟩ in direction V using the ambient product rule. The normal components of D_VW and D_VZ are orthogonal to tangent fields, so V⟨W,Z⟩=⟨∇_VW,Z⟩+⟨W,∇_VZ⟩. In a coordinate basis, equality of mixed derivatives gives Γ^k_(ij)=Γ^k_(ji), the torsion-free condition.

Write g_ij=⟨X_i,X_j⟩. Compatibility gives ∂i g_jk=Γ^ℓ_(ij)g_ℓk+Γ^ℓ_(ik)g_jℓ. Add the equations with i,j exchanged and subtract the one differentiated in k. Symmetry cancels terms, leaving 2g_kℓ Γ^ℓ_(ij)=∂i g_jk+∂j g_ik−∂k g_ij. Multiplication by the inverse metric yields the Christoffel formula. Thus the connection is computed entirely from g and its first derivatives.

For polar coordinates, g is diagonal with entries one and r². Its only nonconstant coefficient has derivative 2r. Substitution gives the table and zero for the remaining symbols.

Apply

For a tangent field with components V^k along a curve with coordinates u^i(t), the covariant derivative has components dV^k/dt+Γ^k_(ij)u′i V^j. The second term is exactly the basis-change correction. Repeated indices denote finite sums over the two coordinates; they are not exponents.

Repair “Christoffel symbols are tensor components.” Their transformation includes second derivatives of coordinate changes. The connection acts geometrically, but its coefficient array is coordinate dependent in a different way from a tensor. Curvature combines derivatives and products of these coefficients so the unwanted coordinate terms cancel.

For optional depth, the metric and torsion-free properties define the Levi-Civita connection on abstract Riemannian surfaces as well. The coefficient derivation supplies its local uniqueness and formula; chart compatibility follows from the geometric transformation law.

Continue

Keep coordinate indices, ordinary derivatives and covariant derivatives distinct in a calculation. Continue to parallel transport and holonomy to use the connection as an equation for moving vectors without intrinsic rotation.