Parallel Transport: Constancy of Magnitudes & Angles Along Geodesic

[latentcorpse]

A vector field Y is parallely propagated (with respect to the Levi-Civita connection)
along an affinely parameterized geodesic with tangent vector X in a Riemannian
manifold. Show that the magnitudes of the vectors X, Y and the angle between
them are constant along the geodesic.

 

[betel]

Some questions to get you started.

What is the definition of parallel transport?
What does along the geodesic mean?
How is the angle between two vectors defined?

 

[latentcorpse]

Some questions to get you started.

What is the definition of parallel transport?
What does along the geodesic mean?
How is the angle between two vectors defined?

The tensor T is parrallely transported along the curve with tangent X^a if \nabla_X T=0

Along the geodesic means along the affinely parameterised curve of shortest distance (think i may be a bit off here but hopefully you can clear it up!)

On a Riemannian manifold, the angle between two vectors is given by

\theta = \cos^{-1} \left( \frac{ g(X,Y) }{ ( |X||Y| ) } \right) where |X|= \sqrt{ g(X,X)}

 

[betel]

So what is your problem?