Curvature of Space in the Context of Cosmology

[Arman777]

Recently I asked a question about the curvature of the universe.

https://www.physicsforums.com/threads/constant-curvature-and-about-its-meaning.977841/

In that context I want to ask something else.

Is this curvature ($\kappa$) different than the Gaussian Curvature ? Like it seems that we are using ##\kappa# for only homogeneous and isotropic spaces.

I guess the crucial point is being "constant " right. The "constant" implies homogeneity and isotropy ?

We can have curvature for any space but only the constant curvature ones will be homogeneous and isotropic ?

 

[timmdeeg]

I guess the crucial point is being "constant " right. The "constant" implies homogeneity and isotropy ?

We can have curvature for any space but only the constant curvature ones will be homogeneous and isotropic ?

Wasn't this already answered in the other thread?

The existence of $\kappa$ at all implies an isotropic homogeneous universe.

I'm not really sure what you mean.

 

[PeterDonis]

Is this curvature ($\kappa$) different than the Gaussian Curvature ?

It is the sign (positive, negative, or zero) of the (constant) Gaussian curvature of a homogeneous and isotropic space. As @Orodruin said in the other thread, this depends on a particular choice of coordinates, in which the sign $\kappa$ is separated from the scaling of distances; the actual Gaussian curvature depends on the scaling of distances as well as the sign.

it seems that we are using $\kappa$ for only homogeneous and isotropic spaces.

Yes, because only homogenous and isotropic spaces have the same Gaussian curvature everywhere, so only those spaces permit the kind of choice of coordinates that allows you to separate the sign of the Gaussian curvature from the scaling, as discussed above.

 

[Arman777]

I see it now, $K = \kappa/R^2$Where ##K# is the Gaussian Curvature.