(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] computing affine connection

I've got a task of computing the components of the Riemann tensor, starting from the given Robertson-Walker metric

[tex]

g=-dt\otimes dt + a^2(t)\Big(\frac{dr\otimes dr}{1-kr^2} + r^2\big(d\theta\otimes d\theta + \sin^2\theta d\phi\otimes d\phi\big)\Big)

[/tex]

When I searched the lecture notes, I found out, that I know how to proceed once the connection coefficients [itex]\Gamma^{\lambda}{}_{\mu\nu}[/itex] are known. So I want to solve what these connection coefficients are. They are defined with the equation

[tex]

\nabla_{e_{\mu}} e_{\nu} = \Gamma^{\lambda}{}_{\mu\nu} e_{\lambda}

[/tex]

where [itex]e_{\mu}[/itex] are some basis of the tangent space in some point on the manifold. The mapping [itex](X,Y)\mapsto \nabla_X Y[/itex] that maps pair of vectors into one vector is an affine connection.

So I searched the lecture notes for the definition of the affine connection. The mapping is defined by postulating that it is linear in both variables, and that if [itex]f[/itex] is some smooth real valued function on the manifold, then

[tex]

\nabla_{fX} Y = f\nabla_X Y

[/tex]

and

[tex]

\nabla_X (fY) = (Xf) Y + f\nabla_X Y.

[/tex]

Having found the definition, I tried to calculate

[tex]

\nabla_{\partial_t} \partial_r \quad\quad\quad\quad\quad\quad\quad (1)

[/tex]

and I was forced to notice that I have no clue of how to do this. I know how the nabla is bilinear, and I know what happens if the tangent vectors get scaled by real valued function, but I don't know what the vector in equation (1) is supposed to be. I was unable to find any clear formula out of the lecture notes that would have told how to actually compute these vectors.

So I want to know, that what am I supposed to start doing, if I want to find out what vector the vector in (1) is. Is there a formula for this?

Or that is what I believe I want to know. If it looks like I started doing the exercise too difficult way, any hints are welcome.

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# Homework Help: Computing affine connection

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