Question: In this exercise we will determine the condition that a vector field can be considered to be globally parallel on a manifold. More precisely, what

In this exercise we will determine the condition that a vector field can be considered to be globally parallel on a manifold. More precisely, what guarantees that we can find a vector field satisfying the equation

(a) A necessary condition, called the integrability condition for this equation, follows from the commuting of partial derivatives. Show that VÎ±,Î½Î² = VÎ±,Î²Î½ implies

(b) By relabeling indices, work this into the form

This turns out to be sufficient, as well.

(VV) = V" = ve +r"ug VH = 0? V" =

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