Introduction to the atlas package

Copyright © 2004-2008  DigiArea Group . All rights reserved.

Description:

• atlas is a Maple™ package for modern differential geometry calculations. It can work with manifolds, mappings, embeddings, submersions, p-forms, tensor fields etc. Thus the user may concentrate on differential geometry problems not on the programming.
• Standard mathematical notations accepted in modern differential geometry  are used in the package. All calculations are as coordinate free as possible; some calculations with symbolic dimension are available.
• In the atlas  package all calculations are performed in terms of tensors, vectors and p-forms. For instance, conformally flat metric tensor of sphere  is presented as , where    are coframe 1-forms and symbol - `&.` is tensor product operator (see examples below). To get more information about differential geometry calculations and to look through some complete examples see atlas[examples] . To look through references list see atlas[references] .
• The atlas  package can make some useful calculations even if  the working dimension is symbolic. For example, if  is the dimension,  are coframe 1-forms and  are frame vectors then decomposition  (of interior product of vector  and 1-form ) - is one of the available calculation. To get more information about this possibility see atlas[dim]
• The atlas  package uses its own data types, called const, scalar, vect, tensor, form, func, domain, mapping, coframe, frame to represent corresponding objects. These types are implemented as procedures (e.g. `type/form`) with a Boolean return value. It returns a value of "true" if its first argument satisfies the properties of a corresponding object, and "false" otherwise. For more information, see atlas[types] .
• Because computations with tensors and p-forms usually involve a great number of quantities, it is important to make simplification in each step of the computations. For this reason, the user can customise the simplification routine `atlas/simp` for a particular problem. For more information, see atlas[simp] .
• In the atlas  package any object (constant, tensor, p-form, manifold etc.) can be indexed. For more information on atlas  indexing facilities, see atlas[indexing] .
• Before using routines of the atlas  package, you must load the package with one of the commands with(atlas) , with(atlas,[]), with(atlas,<function>)  or atlas[<function>] . See with  for details.
• The procedures available are:
   

Examples:
restart:
with(atlas):

Conformally flat metric on sphere:

Declare constant :
Constants(lambda);

Declare functions:
Functions(f=f(x,y),h=h(f));

Declear forms:
Forms(e[i]=1,xi=1,theta=p);

Declare vectors:
Vectors(E[j],X,Y,Z);

Declare coframe 1-forms:
Coframe(e[1]=d(x),e[2]=d(y));

Now coframe 1-forms are .

Declare frame vectors:
Frame(E[j]);

Now frame vectors are .

Metric declaration:
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y))/(1+lambda*(x^2+y^2))^2);

Calculate connection 1-forms:
Connection(omega);

Calculate curvature 2-forms:
Curvature(Omega);

Riemann tensor:
Riemann(R);

Ricci tensor:
Ricci(r);

Ricci scalar:
RicciScalar(s);

Show 1-form :
'omega[2,1]'=factor(omega[2,1]);

Verify that there is no Killing vector field among frame vector fields:
'L[E[j]](g)'=L[E[j]](g);

Verify that "rotation" vector field   is a Killing one:
Z:=x*E[2]-y*E[1];
'L[Z]'(g)=L[Z](g);

Calculation of volume form using Hodge operator &** :
'&**(1)'=radsimp(&**(1));

Some more calculations:

Using interior product operator - :
'iota[E[j]](&**(1))'=iota[E[j]](radsimp(&**(1)));

The same with metric tensor g:
'iota[E[k]](g)'=iota[E[k]](g);

Using exterior derivative operator - :
'd(f)'=d(f);

'd(h)'=d(h);

'd(f*xi)'=d(f*xi);

'&**(d(f))'=radsimp(&**(d(f)));

As  are coframe 1-forms and  are frame vectors then:
'iota[E[i]](e[k])'=iota[E[i]](e[k]);

"ToBasis" decomposition:
X=ToBasis(X);
xi=ToBasis(xi);

See Also:

atlas[types] , atlas[simp] , atlas[dim]  , atlas[examples] , atlas[references] .