Proving the Relation Between Weyl Tensor, Ricci Tensor & Scalar

[sroeyz]

Hello, I wish to show that on 3-dimensional manifolds, the weyl tensor vanishes.
In other words, I want to show that the curvature tensor, the ricci tensor and curvature scalar hold the relation

Please, if anyone knows how I can prove this relation or refer to a place which proves the relation, I will be most grateful.

Thanks in advance

 

[jim mcnamara]

You have created an identical thread in the Physics area. Please don't double post.

 

[stephen_weber]

Hey! Jim McNamara ...

Excuses Excuses Jim McNamara.

I was interested in his question. I find it obnoxious that someone that didn't even participate in providing him with an answer in the physics section has the audacity to chid him for being interested in asking a larger audience.

In another person's post about unitary matrices you tried answering with some nonsense, that Matt grime cleaned up.

In a post called complex.h . You again gave bogus statements that were cleaned up by Hurkyl.

In a post Atomic number and Orbitals. You again make a God like statement about passing on answering it as you think it is someone's homework and it is incomprehensible. But others gave him a clear answer.

In Asymptotic mathcing for a first order differential equation post . You again declare that a variable "e" in some equation must refer to Napier's constant. At least there you start with "I'M CONFUSED".

In Sulfur Based Lifeforms Question post . You don't even get that the point is that we are carbon based life forms.

I am not in charge of this forum. But please don't TELL anyone else anything, ok. (Especially about math, it isn't based on an opinion which you obviously want to flaunt).

Steve

 

[Doodle Bob]

the long and short of the proof is: first show that the tensor is skew-symmetric in any two variables; then take a basis of TM and plug them into the weyl tensor; the weyl tensor is a 4-tensor, so one of the basis elements has to double up; hence it's zero.