Understanding Ricci and K Curvature in 2 Dimensions: A Simple Explanation

[stephen_weber]

Hi,

In two dimensions I am under the impression that the ricci tensor or the scalar curvature equals the negative of the fundamental tensor and the sectional curvature (K).
I'd have written it out with the proper symbols but I am new to this forum and this isn't at least a complex question.

I know that the sectional curvature is directly proportional to the Riemannian Tensor, and since I am only talking about two dimensions, the only term that is independent and nonzero is R 1212. OK with the symmetries there are dependent terms that are the positive and negative of that but all of the multiplicities cancel out in the definition of K.

I was wondering if there where was anyone out there who can walk me or US through how this equation is true?

 

[pervect]

What do you want a walkthrough of, exactly? It sounds like you already know everything.

 

[stephen_weber]

Thank you for that comment. I am trying to make all these curvature equations mean something to me so that I can understand General Relativity.
To be dead honest I am working on a problem in Schaum's "Tensor Calculus", chapter eight . problem 8.30(a). Without special symbols here if the characters in brackets are subscripts then the question is simply.
-------------------------------------
Show that in a Riemannian 2-space...
R[ij] = - g[ij]*K
--------------------------------------
I have followed through this chapter meticulously. So I don't need help understanding certain points about all this like how the symmetries in R make some terms dependent on others. And one of the earlier problems dealt with K in 2-space in which (problem 8.7,8.6) K was shown to reduce to

K= R[1212]/g

or

K= R[1212] / ( g[11]*g[22]-g[12]^2 )



Without me drowning on and on how do I show that ?

 

[stephen_weber]

I posted this in Tensor Math and after a few days worked out the answer with some help from Doodle Bob.
With one detail change that schaum's book has the symmetry inverted or the contraction on the last term instead of the middle, this only effects the negative sign in the answer. So here is the proof.



Fact One : In n=2 ::: K=R[1212]/g where g=g11g22-g12g21

Desired Result ::: R[ij] = g[ij]*K (noting that the original negative is based on direction of curvature and Schaum's is in the minority )

Starting with::: R[ij] = R[ikj]^[k] = g^[hk]* R[hikj]
and using
g^[hk]= g[hk]^-1
as a substitution
R[ij] = g[hk]^-1 * R[hikj]

Some shortcuts:::: the only non zero R's are
R[1212]=R[2121]= -R[1221] = -R[2112]
and
g[11]^-1 = g22/g
g[22]^-1 = g11/g
g[12]^-1 = -g12/g
g[21]^-1 = -g21/g
where again g is the determinant g=g11*g22-g12g21

Writting out all the terms for the right hand side of
R[ij]= g[hk]^-1 * R[hikj]

R[ij]=g[11]^-1*R[1212]+g[22]^-1*R[2121]+g[12]^-1*R[1221]+g[21]^-1*R[2112]
All other Summation factors of R equal zero.

substitution of inverses and converting all the R terms to R[1212] gives

R[ij] = R[1212]*(g22/g+g11/g+g12/g+g21/g)

R[ij] = (R[1212]/g)*(g11+g22+g12+g21)

R[ij] = g[ij] * (R[1212]/g)

with fact one being K=R[1212]/g

I have my desired result

R[ij]=g[ij]*K