Riemann Christoffel, cyclic properties

Kiwi_Dave · #1 $Old$ June 17th, 2009, 04:50 AM

Verify that:

$R_{nijk}+R_{njki}+R_{nkij}=0$

Now it is not clear if I may assume that the space is three dimensional. If I can make this assumption then I can verify the equation (see below).

MY QUESTION: Is it possible to verify this equation for a space with more than three dimensions?

When there are only three dimensions one of the tensors is zero because two equal indices occupy either the first and second or third and fourth index. Say n=i, then the first tensor is zero and it remains to prove:
$R_{ijki}+R_{ikij}=R_{ijki}+R_{ijik}=R_{ijki}-R_{ijki}=0$

Rebesques · #2 $Old$ June 23rd, 2009, 09:22 AM

Quote:

MY QUESTION: Is it possible to verify this equation for a space with more than three dimensions?

Ofcourse. This is the famous Bianchi identity.

Kiwi_Dave · #3 $Old$ June 28th, 2009, 02:10 AM

The Bianchi identity seems to involve covariant differentiation of the Reimann Christoffel tensor. In contrast this question does not.

Or am I missing something?

HallsofIvy · #4 $Old$ June 29th, 2009, 11:02 AM

There are two (related) identitites that are commonly referred to as "Bianchi identities".
Bianchi Identities -- from Wolfram MathWorld