Sage Manifolds

• Graphical output for charts (method Chart.plot()) and points (method Point.plot()); here are some examples.
• Introduction of index notation to denote tensor contractions and tensor symmetrizations (new class TensorWithIndices); see these links: 1, 2.
• The argument of methods symmetrize() and antisymmetrize() in tensor classes is now directly a sequence of index positions (and no longer a single list/tuple encapsulating such a sequence).
• Method self_contract() of tensor classes renamed trace().
• The code for tensor contractions has been optimized; moreover multiple tensor contractions are now allowed.
• The documentation (reference manuals 4 and 5) has been improved.

• the introduction of curves in manifolds (with some plotting capabilities)
• improvements in differential mappings between manifolds, including mapping composition and mapping differential
• the introduction of homomorphisms between free modules

• Plot of vector fields: new method VectorField.plot()
• Possibility of parallelizing heavy computations: parallelization is implemented for basic tensor calculus (arithmetics, contractions) and for calculus regarding affine connections (connection coefficients, action on a tensor field, Riemann curvature tensor)
• Nice display of partial derivatives
• Standard math functions exp, cos, sin, etc. on scalar fields
• Display of tensor components as a list, one per line: new methods TensorField.display_comp() and FreeModuleTensor.display_comp()
• Nice display of connection coefficients: new method AffConnection.display()
• Nice display of Christoffel symbols: new method Metric.christoffel_symbols_display()
• Nice display of chart transition maps: new method CoordChange.display()

• Topological properties have been separated from differential ones, by implementing topological manifolds (new classTopologicalManifold) and making the class for differentiable manifolds (DifferentiableManifold) inherit from TopologicalManifold.
• The base field over which manifolds are defined is no longer assumed to be the real field: it can be any topological field (non-discrete to define differentiability for diff. manifolds). This allows to define easily complex manifolds, by setting the field to C.
• The class ManifoldOpenSubset has been suppressed: open subsets of manifolds are now instances of TopologicalManifold or DifferentiableManifold (since an open subset of a top/diff manifold is a top/diff manifold by itself)
• Functions defined on a coordinate patch are no longer necessarily symbolic functions of the coordinates: they now pertain to the generic class CoordFunction, symbolic functions being described by a subclass of it (CoordFunctionSymb). This opens the way for "numerical" manifolds, like spacetimes generated by numerical relativity codes.
• Better parallelization, governed by the new singleton class Parallelism and the global function use_multiproc.

This release propagates further code changes related to the integration of SageManifolds into SageMath (cf. the metaticket#18528); it also adds a few new functionalities.

New functionalities:

• Computation of the Schouten tensor, the Cotton conformal tensor and the Cotton-York conformal tensor associated to a given pseudo-Riemannian metric
• Add structure of Lie algebroid to modules of vector fields (classes VectorFieldModule and VectorFieldFreeModule): new method VectorField.bracket
• Parallelization of vector field plots
• Parallelization of arithmetics of fully antisymmetric tensor components
• Improved rendering of variables in partial derivatives, using LaTeX display of symbols (class ExpressionNice)
• Add comparison operator for transition maps (class CoordChange)
• Add list functionalities for bases (methods __len__ and __iter__ in classes FreeModuleBasis and FreeModuleCoBasis)

Syntactic changes:

• Method CoordChange.set_inverse: replace the keyword check by verbose, the default being now verbose=False
• Introduction of Manifold.options to control the display of mathematical expressions instead of the global functions nice_derivatives and omit_function_args, which have been suppressed
• Function set_axes_labels (to set labels on 3D plots) no longer imported at the startup time; if required, one has to type from sage.manifolds.utilities import set_axes_labels
• Function xder (exterior derivative) no longer imported at the startup time; if required, one has to type from sage.manifolds.utilities import xder
• Class DiffForm: method exterior_der renamed exterior_derivative
• Classes DiffScalarField, TensorField and TensorFieldParal: method lie_der renamed lie_derivative, with lie_der kept as an alias of the latter

More internal changes:

• Manifold structure now described via specific singleton classes: TopologicalStructure, RealTopologicalStructure, DifferentialStructure and RealDifferentialStructure
• Class TopologicalManifoldSubset renamed ManifoldSubset
• Class TopologicalManifoldPoint renamed ManifoldPoint
• Manifold subsets are no longer facade parents
• Class ManifoldSubset: new methods lift and retract
• Introduction of the commutative algebra of all symbolic coordinate functions on a given chart: new class CoordFunctionSymbRing and class CoordFunction now inheritates from AlgebraElement
• Class FiniteRankFreeModule: category changed from Modules(ring) to Modules(ring).FiniteDimensional()
• Some changes to prepare the migration to Python 3 (e.g. print replaced by print())

Syntactic changes:

• Method plot of classes RealChart and VectorField: keyword argument nb_values renamed number_values
• Method structure_coef of class VectorFrame renamed structure_coeff
• Class OpenInterval: argument subinterval_of renamed ambient in the constructor
• Class RealLine: LateX name changed from \RR to \Bold{R}

Internal changes:

• Systematic use of Python3-compatible syntax (to prepare the migration of SageMath to Python3); in particular:
  • All occurrences of iteritems() changed to items()
  • All occurrences of itervalues() changed to values()
  • Classes ScalarField and TensorField: method __nonzero__ renamed __bool__
  • Class TensorField: method __div__ renamed __truediv__
• Classes TensorFieldModule, VectorFieldModule and DiffFormModule: add cached method zero
• Classes DiffForm and DiffFormParal: method exterior_derivative is cached (via the decorator @cached_method)
• Class VectorFrame: method structure_coeff is cached (via the decorator @cached_method)

Other changes:

• Improvements in the documentation; in particular 3D graphics have been added to the reference manual for illustrating the use of some plot methods
• Values set by the user to some keyword arguments of plot methods become the new default values until further explicit change (this behavior is provided by the decorator @options):
  • TangentVector.plot: argument scale
  • VectorField.plot: arguments max_range, scale and color
  • DifferentiableCurve.plot: arguments thickness, plot_points, max_range and aspect_ratio