For those users that followed my post of past August 29, I must say that I got a response by a person (by name *Lark 49*) from Maplesoft [to my suspects of a serious bug in MAPLE 13] which response contained ANOTHER way of computing the Riemann tensor, the Ricci tensor and the Riemann scalar curvature, this time using Maple-13's other Package [Differential Geometry] that this time returned a correct response to my quests and serious doubts.
The fact is: there are two built-in Packages of Maple-13 that return TWO different responses to a very concrete problem of TensorCalculus and/or Differential Geometry & it is plain even to an ass that one of these two responses is TRUE & the other is FALSE because they differ in their signs. I did use the two procedures in a few other instances of 3-dimensional & 4-dimensional Manifolds and I came to verify that the two procedures keep returning different signs in the Riemann scalar curvature of those manifolds !!
So the point is: if I get 2 different and incompatible responses to the same mathematical problem, how can I ever BE SURE of the correctness of the 2nd procedure (Differential Geometry) over the 1st (Tensor Package), in cases I have not calculated by hand the responses? The same thing (TWO contraditory responses) keeps repeating over & over again and I have lost my confidence in the Maple-13 software !!
So, is the Differential Geometry Package always right & the Tensor Package always wrong? -- as I verified to be the case for a few cases -- or can there be instances when it is the former which fails & the latter is the correct one?