## Modern differential geometry

Modern differential geometry is the basis for the package. The entities such as manifolds, mappings, p-forms, tensor fields, bundles, connections are very important in the modern differential geometry. The package allows you to work with these entities without extra efforts. Define an entity with the corresponding obvious definition and work with it just as you usually do.

- Domain - manifold and domain declaration
- Constants - constants declaration
- Functions - functions declaration
- Tensors - tensors declaration
- Forms - forms declaration
- Vectors - vectors declaration
- Mapping - declaration of a mapping between manifolds or domains
- Coframe - coframe declaration
- Frame - frame declaration
- Metric - metric tensor declaration

## No programming just differential geometry

When working on your problem you think in terms of manifolds, mappings, embeddings, submersions, p-forms, tensor fields
etc.

The
package allows you to concentrate on differential geometry problems, but not on the programming.

You can use predefined declaration operators to declare various differential geometry objects, which are calculated on the fly:

- Projectors - automatic calculation of projectors of a mapping
- Invariants - automatic calculation of invariants of a mapping
- Connection - automatic calculation of connection 1-forms
- Curvature - automatic calculation of curvature 2-forms
- Torsion - automatic calculation of torsion 2-forms
- Riemann - automatic Riemann tensor calculation
- Ricci - automatic Ricci tensor calculation
- RicciScalar - automatic Ricci scalar calculation

## No ugly output just standard notations

The Atlas 2 package uses standard differential geometry notations: d - exterior derivative, - Lie derivative, ι - interior product, - exterior product, - tensor product, - Hodge star, - covariant differentiation, δ - Kronecker's delta symbol etc. You always get output as you expected like the following:

- Output example with Lie derivative calculation:
- Output example with exterior derivative calculation:
- Output example with tensor product calculation:
- Output example with covariant derivative calculation:
- Output example with interior product and Kronecker's delta symbol calculation:
- Output example with calculation in a manifold with symbolic dimension:

## Single solving path for almost any problem

With the Atlas package you always have one and the same solving path for almost all your differential geometry problems. You start with definitions of manifolds, vector and tensor fields, p- forms and mappings between the manifolds.

When you get your differential geometry entities defined, you use standard operators to get various quantities of your entities:

- Projectors - automatic calculation of projectors of a mapping
- Invariants - automatic calculation of invariants of a mapping
- Connection - automatic calculation of connection 1-forms
- Curvature - automatic calculation of curvature 2-forms
- Torsion - automatic calculation of torsion 2-forms
- Riemann - automatic Riemann tensor calculation
- Ricci - automatic Ricci tensor calculation
- RicciScalar - automatic Ricci scalar calculation

This is standard procedure which can be automated completely.

## All calculations are as coordinate free as possible

In the package all calculations are performed in terms of tensors, vectors and p-forms (not their components!). 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 the main principle of the package structure and to look through some complete examples see Examples.

To look through references list see References.

## Some calculations with symbolic dimension are available

The package allows one to 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. Another example is Lie bracket decomposition:

To get more information about this possibility see Dimension.

## Almost any differential geometry entity can be indexed

In the package any object (constant, tensor, p-form, manifold etc.) can be indexed. This is very flexible feature. For or more information on indexing facilities, see Indexing.

## Easy customizable simplification of your results

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 customize the simplification routine `atlas/simp` for a particular problem. For more information, see Simplification routine.

### Library of predefined

differential geometry objects

##### Over 580 differential geometry objects make Atlas 2 more powerful

Now your work can be enriched by:

### DG Library Browser

##### Advanced Maple Maplet

Now you can generate a Maple worksheet for each of the library objects. The worksheet automatically calculates differential geometry quantities for this entity.

### Atlas 2D/3D Wizard

##### Powerful GUI Add-On for Code Generation

- - calculation of curvature, torsion, tangent, principal normal and binormal vectors for plane and space curves in any coordinate system.
- - calculation of metric, second fundamental form, mean curvature vectors, Laplace operator, connection, curvature Riemann and Ricci tensor, Gauss curvature for any surface in any 3D coordinate system.
- - calculation of metric, connection, Laplace operator for any 2D and 3D coordinate system.