A vector v is parallel transported along a curve γ(t) on a Riemannian manifold. The parallel transport equation is ∇_{γ'(t)} V = 0, where V(t) is the vector field along γ. What does this equation say geometrically?
AThe vector V does not change length along γ
BThe vector V has zero covariant derivative along γ — it is as constant as the connection allows
CThe vector V points in the direction of γ'(t) at all times
DThe vector V is orthogonal to the curve at all times
The equation ∇_{γ'} V = 0 says the covariant rate of change of V along the curve is zero — V is 'not changing' in the sense defined by the connection. On a Riemannian manifold with the Levi-Civita connection, parallel transport also preserves inner products (lengths and angles), but this is a consequence of metric compatibility, not the definition of parallel transport. A vector being parallel does not mean it points along or perpendicular to the curve.
Question 2 True / False
On the 2-sphere with the round metric, parallel transport of a tangent vector around a great-circle triangle with three right angles (an octant boundary) rotates the vector by 90 degrees.
TTrue
FFalse
Answer: True
This is the classic demonstration of curvature via parallel transport. Start at the north pole pointing along a meridian. Transport along the equator (the vector, initially pointing south, stays pointing south — it is parallel on the equator). Turn 90° and go back to the north pole. The vector arrives rotated 90° from its original direction. The angle of rotation equals the integral of Gaussian curvature over the enclosed region: for a sphere of radius 1, K = 1, area of the octant = π/2, and the rotation angle is π/2 = 90°. The Gauss-Bonnet theorem generalizes this.
Question 3 Short Answer
On a flat manifold (zero curvature), parallel transport around any closed loop returns a vector to its original value. Why?
Think about your answer, then reveal below.
Model answer: Zero curvature means the connection is locally the standard flat connection, where Christoffel symbols can be made to vanish in suitable coordinates (normal coordinates). In flat coordinates, parallel transport is just ordinary constant-vector transport, which is trivially path-independent. More precisely, zero curvature implies the holonomy group is trivial — the parallel transport around any contractible loop is the identity map on the tangent space. Path-independence of parallel transport is equivalent to vanishing curvature.
The Ambrose-Singer theorem formalizes this: the Lie algebra of the holonomy group is generated by curvature tensors. Zero curvature gives a trivial holonomy Lie algebra, hence trivial holonomy for contractible loops. On a manifold with nontrivial fundamental group (like a flat torus), parallel transport around non-contractible loops can be nontrivial even with zero curvature — this is the distinction between local and global holonomy.
Question 4 True / False
Parallel transport along a curve preserves the Riemannian inner product of two parallel vector fields: d/dt g(V, W) = 0 when ∇_{γ'} V = 0 and ∇_{γ'} W = 0.
TTrue
FFalse
Answer: True
This follows from metric compatibility of the Levi-Civita connection: d/dt g(V,W) = g(∇_{γ'} V, W) + g(V, ∇_{γ'} W) = 0 + 0 = 0. Parallel transport is therefore an isometry between tangent spaces — it preserves lengths, angles, and the inner product. This is a special property of the Levi-Civita connection; a non-metric connection's parallel transport would not preserve the inner product.