General relativity: constant curvature, characterizing equation

[Derivator]

Homework Statement

Show, that a three-dimensional space with constant curvature K is charaterized by the following equation for the Riemann curvature tensor:

$$R_{abcd} = K \cdot \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right)$$

Homework Equations

The Attempt at a Solution

Hi folks,

I would like to give an own attempt, but I have no Idea how to start.

We haven't defined the curvature K in lecture. How is it defined?
Has anybody an idea, how to start?

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derivator

 

[Derivator]

anybody an idea?

 

[gabbagabbahey]

Wald defines the Ricci tensor as

$$R_{ac}=R_{abc}{}^{b}$$

And then the scalar curvature is the trace of the Ricci tensor

$$R=R_{a}{}^{a}$$

Your text should have similar definitions.

So, I think want you want to do is to show that if $R_{abcd} = K\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right)$, where $K$ is a constant, then the scalar curvature is $R=K$.

 

[Derivator]

ah, thanks for your input.

i haven't seen this exercise from this point of view, but it makes sense.

I have found: R=6K, thus R is constant, thus we have a constant curvature.