General Relativity: An Introduction for Physicists · Problem 3.13

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · parallel-transport-sphere

Problem

On the surface of a unit sphere,

ds^2 = d\theta^2 + \sin^2\theta\, d\phi^2

. Calculate the connection coefficients in the

(\theta, \phi)

coordinate system. A vector

\boldsymbol{v}

of unit length is defined at the point

(\theta_0, 0)

as parallel to the circle

\phi = 0

. Calculate the components of

\boldsymbol{v}

after it has been parallel-transported around the circle

\theta = \theta_0

. Hence show that, in general, after parallel transport the direction of

\boldsymbol{v}

is different but its length is unchanged.