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Fredrik
#16
Mar29-09, 12:24 PM
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One of the conditions on [itex]\nabla[/itex] mentioned in the definition of a connection is that

[tex]\nabla_X(fY)=(Xf)Y+f\nabla_XY[/tex]

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

[tex]\nabla_Xf=Xf[/tex]

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

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

so

[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

[tex]\nabla_X(fY)=(\nabla_Xf)Y+f\nabla_XY[/tex]

The other two requirements in the definition of a connection are

[tex]\nabla_X(Y+Z)=\nabla_XY+\nabla_XZ[/tex]

[tex]\nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ[/tex]