The Riemann curvature tensor was defined in an earlier blog post, e.g.
as: R n b m n
This tensor , as noted in that post, is of key importance in general relativity. The Ricci tensor of the first kind is defined simply as a contraction of the Riemann tensor of the 2nd kind, i.e. the above curvature tensor. Thus:
R b m = R n b m n
Raising an index yields the Ricci tensor of the second kind, e.g. R b m
It is completely determined by knowing the quantity R b m for all vectors V i of unit length. The tensor is obtained by defining a Ricci tensor of the 2nd kind thus:
(Note: Value of g 22 is not the same as g 22 !)
* Assume a metric given by the 'g' values:
Example (1): Find: g 11 G 2 21
Example (2): Find: g 22 G 3 32
Recall the relations of Riemann tensors to Christoffel values, e.g.
R 1 212 = - 1 - G 1 22 G 1 11 - G 2 21 G 1 22 = 0