View Single Post
Mar29-09, 12:24 PM
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,415
One of the conditions on [itex]\nabla[/itex] mentioned in the definition of a connection is that


Note that Xf and fX are not the same. Xf is the function [itex]p\mapsto X_pf[/itex] and fX is the vector field [itex]p\mapsto(p,f(p)X_p)[/itex]. The effect of [itex]\nabla_X[/itex] on a function is defined as


Use this definition with [itex]X=\partial_\mu[/itex], and you get

[tex]\nabla_{\partial_\mu}f=\partial_\mu f[/tex]


[tex]t(f)=t^\mu\partial_\mu f=t^\mu\nabla_{\partial_\mu}f[/tex]

Note also that the definition allows us to rewrite the first equation as


The other two requirements in the definition of a connection are