432 10 Conclusion of Volume 2

4. doricci

This command generates as an output a tensor ricten[[a,b]] =Rcbca where Rcbca is the Ricci tensor in the conventions of the old Kaluza-Klein literature, namely Universal mass relations Ann. of Phys. 162, (1985) 372 by D’Auria and Frè.

5. docurvaform

This command generates the curvature 2-form once the Riemann tensor has been generated

Index ordering

The index ordering is as follows:

A=(a,i); a=1,.....,dim G/H; i=dim G/H+1,........,dim G

The index a enumerates the coset directions, while the index i enumerates the H subalgebra directions.

Structure Constants for CP2

This programme calculates the generators and the structure constants of SU(3) in

such a way that the first 4 generators are those of the coset:

SU(3)

SU(2)×U (1)

Spheres

This programme calculates the generators and the structure constants of the group SO(n + 1) and orders them in such a way that the first n generators are those of the

coset:

SO(n+1)

SO(n)

N010 Coset

This programme generates the structure constants of SU(3) but in such an order that

the first 7 generators are those of the coset

SU(3)

U (1)

U(1) being generated by the 8th Gell Mann matrix

RUNCOSET Package (Euclidian Signature)

This is a new package based on the package “Cosets” that was written by Leonardo Castellani and which computes the Riemann tensor, Weyl tensor and the spin connection for G/H manifolds. The package has been modified and adapted to the way we want to perform our calculations.

Geometry of Quasi-Homogeneous Manifolds (Euclidian,

or Lorentzian) and of General Manifolds (General Signature)

This routine is devised to calculate the geometry in the following very common situation where the vielbein of a space in dimension

n=1+r

is given in the following form: e1=dμ

ei =fi (μ)i σ i

where fi (μ) are functions of the coordinate μ and σ i are vielbeins of an r-

dimensional space for which the contorsion is already known:

dσ i = - tjki σ j σ k

We name such manifolds quasi-homogeneous

Main

You start this programme by typing mainspin

Spin Connection and Curvature Routines

This routine is devised to calculate the intrinsic components of the spin connection

once the contorsion tensor as already been calculated

dei = cjki ei ek

There are two versions of the programme one for quasi-homogeneous manifolds called spinpack and one for general manifolds called spinpackgen in the second case you will be prompted to supply also the signature of space-time a n vector of plus or minus 1s = signat

Routine Curvapack

This routine is devised to calculate the curvature of a manifold when the intrinsic components of the spin connection depend only on one coordinate μ and the first vielbein is

e1= A(μ)*dμ

Routine Curvapackgen

This routine is devised to calculate the curvature two form and the Riemann tensor in a general situation for an arbitrary dimensional manifold and with the vielbein depending on all the coordinates. You can start this programme only after having computed the spin connection via the package spinpack