B Question regarding if curvature is not constant

[Apashanka]

For 2-sphere it is having a curvature of k=1/R ,where R is the radius of the 2-sphere and to make it more generalised we treat the kR as the curvature which is always +1 and is independent of it's radius.
My question is how do we treat the curvature term for 3-sphere ,
And it the curvature term is not constt. unlike the 2-sphere ,in general how do we treat those cases and how it's functional form is determined??

Thank you

 

[PeterDonis]

Curvature in general is not a scalar, it's a tensor, the Riemann curvature tensor. It just happens that for a 2-D surface the curvature tensor has only one independent component, so it's the same as a scalar. But for a 3-D manifold, the curvature tensor has 6 independent components, and for a 4-D manifold (like spacetime in General Relativity) it has 20 independent components.

Manifolds with constant curvature are manifolds where all of the curvature tensor components are constant, i.e., the same everywhere.

 

[Apashanka]

Curvature in general is not a scalar, it's a tensor, the Riemann curvature tensor. It just happens that for a 2-D surface the curvature tensor has only one independent component, so it's the same as a scalar. But for a 3-D manifold, the curvature tensor has 6 independent components, and for a 4-D manifold (like spacetime in General Relativity) it has 20 independent components.

Manifolds with constant curvature are manifolds where all of the curvature tensor components are constant, i.e., the same everywhere.

Ok that means that for flat 2-D surface and for flat 3-D surface the Reimann curvature tensor components is 0 but for 2-sphere and 3-sphere the Reimann curvature tensor components are not all zero ,that's how it is determined that the space is flat or curved??

 

[PeterDonis]

that means that for flat 2-D surface and for flat 3-D surface the Reimann curvature tensor components is 0 but for 2-sphere and 3-sphere the Reimann curvature tensor components are not all zero ,that's how it is determined that the space is flat or curved??

That's correct, "flat" means all components of the Riemann tensor are zero, "curved" means they're not.

 

[Apashanka]

That's correct, "flat" means all components of the Riemann tensor are zero, "curved" means they're not.

Ok sir thanks a lot.
That means sir Reimann tensor is the parameter which is used to know a surface is flat or curved.
But sir flat or curved aren't they are relative term (e.g flatness or curvedness is measured with respect to some thing)??
Am I right sir??

 

[PeterDonis]

flat or curved aren't they are relative term

No.

 

[PeterDonis]

will you please explain elaborately

No. What is hard to understand about my answer? If you think the answer should be different than the one I gave, then you should explain why you think that.