Cartan.jl

TensorField topology over FrameBundle ∇ with Grassmann.jl elements

Cartan.jl introduces a pioneering unified numerical framework for comprehensive differential geometric algebra, purpose-built for the formulation and solution of partial differential equations on manifolds with non-trivial topological structure and Grassmann.jl algebra. Written in Julia, Cartan.jl unifies differential geometry, geometric algebra, and tensor calculus with support for fiber product topology; enabling directly executable generalized treatment of geometric PDEs over grids, meshes, and simplicial decompositions.

The system supports intrinsic formulations of differential operators (including the exterior derivative, codifferential, Lie derivative, interior product, and Hodge star) using a coordinate-free algebraic language grounded in Grassmann-Cartan multivector theory. Its core architecture accomodates numerical representations of fiber bundles, product manifolds, and submanifold immersion, providing native support for PDE models defined on structured or unstructured domains.

Cartan.jl integrates naturally with simplex-based finite element exterior calculus, allowing for geometrical discretizations of field theories and conservation laws. With its synthesis of symbolic abstraction and numerical execution, Cartan.jl empowers researchers to develop PDE models that are simultaneously founded in differential geometry, algebraically consistent, and computationally expressive, opening new directions for scientific computing at the interface of geometry, algebra, and analysis.

 _________                __                  __________
 \_   ___ \_____ ________/  |______    ____   \\       /
 /    \  \/\__  \\_  __ \   __\__  \  /    \   \\     /
 \     \____/ __ \|  | \/|  |  / __ \|   |  \   \\   /
  \______  (____  /__|   |__| (____  /___|  /    \\ /
         \/     \/                 \/     \/      \/

developed by chakravala with Grassmann.jl

Tensor field topology and fiber bundles

Provides TensorField{B,F,N} <: GlobalFiber{LocalTensor{B,F},N} implementation for both a local ProductSpace and general ImmersedTopology specifications on any FrameBundle expressed with Grassmann.jl algebra. Many of these modular methods can work on input meshes or product topologies of any dimension, although there are some methods which are specialized. Building on this, Cartan provides an algebra for FiberBundle sections and associated bundles on a manifold, such as general Connection, LieDerivative, and CovariantDerivative operators in terms of Grassmann elements. Calculus of Variation fields can also be generated with the combined topology of a FiberProductBundle. Furthermore, the FiberProduct structure enables construction of HomotopyBundle types. Utility package for differential geometry and tensor calculus intended for Adapode.jl.

Definition. Commonly used fundamental building blocks are

• ProductSpace{V,K,N} <: AbstractArray{Chain{V,1,K,N},N}
  • uses Cartesian products of interval subsets of real line products
  • generates lazy array of Chain{V,1} point vectors from input ranges
• Global{N,T} represents array with same T value at all indices
• LocalFiber{B,F} has a local basetype of B and fibertype of F
  • Coordinate{P,G} has pointtype of P and metrictype of G
• ImmersedTopology{N,M} = AbstractArray{Values{N,Int},M}
  • ProductTopology generates basic product topologies for grids
  • SimplexTopology defines continuous simplex immersion
  • DiscontinuousTopology disconnects for discontinuous
  • LagrangeTopology extends for Lagrange polynomial base
  • QuotientTopology defines classes of quotient identification

Generalizing upon ProductTopology, the QuotientTopology defines a quotient identification across the boundary fluxes of the region, which then enables different specializations of QuotientTopology as

• OpenTopology: all boundaries don't have accumulation points,
• CompactTopology: all points have a neighborhood topology,
• CylinderTopology: closed ribbon with two edge open endings,
• MobiusTopology: twisted ribbon with one edge open ending,
• WingTopology: upper and lower surface topology of wing,
• MirrorTopology: reflection boundary along mirror (basis) axis,
• ClampedTopology: each boundary face is reflected to be compact,
• TorusTopology: generalized compact torus up to 5 dimensions,
• HopfTopology: compact topology of the Hopf fibration in 3D,
• KleinTopology: compact topology of the Klein bottle domain,
• PolarTopology: polar compactification with open edge boundary,
• SphereTopology: generalized mathematical sphere, compactified,
• GeographicTopology: axis swapped from SphereTopology in 2D.

Combination of PointArray <: Coordinates and ImmersedTopology leads into definition of TensorField as a global section of a FrameBundle.

Utility methods for QuotientTopology include isopen and iscompact, while utility methods for SimplexTopology include nodes, sdims, subelements, subimmersion, topology, totalelements, totalnodes, vertices, elements, isfull, istotal, iscover, isdiscontinuous, isdisconnected, continuous, disconnect, getfacet, getimage, edges, facets, complement, interior, ∂, degrees, weights, adjacency, antiadjacency, incidence, laplacian, neighbors.

Definition. A fiber bundle is a manifold E having projection which commutes with local trivializations paired to neighborhoods of manifold B, where B is the basetype and F is the fibertype of E.

FiberBundle{E,N} <: AbstractArray{E,N} where E is the eltype

• Coordinates{P,G,N} <: FiberBundle{Coordinate{P,G},N}
  • PointArray{P,G,N} has pointtype of P, metrictype of G
  • FiberProduct introduces fiber product structure for manifolds
• FrameBundle{C,N} has coordinatetype of C and immersion
  • GridBundle{N,C} N-grid with coordianates and immersion
  • SimplexBundle{N,C} defines coordinates and an immersion
  • FaceBundle{N,C} defines element faces and their immersion
  • FiberProductBundle for extruding dimensions from simplices
  • HomotopyBundle encapsulates a variation as FrameBundle
• TensorField defines fibers in a global section of a FrameBundle

When a TensorField has a fibertype from <:TensorGraded{V,g}, then it is a grade g differential form. In general the TensorField type can deal with more abstract fibertype varieties than only those used for differential forms, as it unifies many different forms of tensor analysis.

By default, the InducedMetric is defined globally in each PointArray, unless a particular metric tensor specification is provided. When the default InducedMetric is invoked, the metric tensor from the TensorAlgebra{V} type is used for the global manifold, instead of the extra allocation to specify metric tensors at each point. FrameBundle then defines local charts along with metric tensor in a PointArray and pairs it with an ImmersedTopology. Then the fiber of a FrameBundle section is a fiber of a TensorField.

These methods relate to FrameBundle and TensorField instances

• coordinates(m::FiberBundle) returns Coordinates data type
• coordinatetype return applies to FiberBundle or LocalFiber
• immersion(m::FiberBundle) returns ImmersedTopology data
• immersiontype return applies to FiberBundle or LocalFiber
• base returns the B element of a LocalFiber{B,F} or FiberBundle
• basetype returns type B of a LocalFiber{B,F} or FiberBundle
• fiber returns the F element of LocalFiber{B,F} or FiberBundle
• fibertype returns the F type of LocalFiber{B,F} or FiberBundle
• points returns AbstractArray{P} data for Coordinates{P,G}
• pointtype is type P of Coordinate{P,G} or Coordinates{P,G}
• metrictensor returns the grade 1 block of the metricextensor
• metricextensor is AbstractArray{G} data for Coordinates{P,G}
• metrictype is type G of Coordinate{P,G} or Coordinates{P,G}
• fullcoordinates returns full FiberBundle{Coordinate{P,G}}
• fullimmersion returns superset ImmersedTopology which isfull
• fulltopology returns composition of topology ∘ fullimmersion
• fullvertices list of vertices associated to the fullimmersion
• fullpoints is full AbstractArray{P} instead of possibly subspace
• fullmetricextensor is full AbstractArray{G} instead of subspace
• isinduced is true if the metrictype is an InducedMetric type
• bundle returns the integer identification of bundle cache

Various interpolation methods are also supported and can be invoked by applying TensorField instances as function evaluations on base manifold or applying some form of resampling method to the manifold topology. Some utility methods include volumes, initmesh, refinemesh, affinehull, affineframe, gradienthat, etc.

For GridBundle initialization it is typical to invoke a combination of ProductSpace and QuotientTopology, while optional Julia packages extend SimplexBundle initialization, such as Meshes.jl, GeometryBasics.jl, Delaunay.jl, QHull.jl, MiniQhull.jl, Triangulate.jl, TetGen.jl, MATLAB.jl, FlowGeometry.jl.

Standard differential geometry methods for curves includes integral, integrate, arclength, tangent, unittangent, speed, normal, unitnormal, curvature, radius, evolute, involute, osculatingplane, unitosculatingplane, binormal, unitbinormal, torsion, frame, unitframe, curvatures, bishopframe, bishopunitframe, bishoppolar, bishop, planecurve, linkmap, and linknumber. Standard differential geometry methods for surfaces includees graph, normal, unitnormal, normalnorm, jacobian, weingarten, sectordet, sectorintegral, sectorintegrate, surfacearea, surfacemetric, surfacemetricdiag, surfaceframe, frame, unitframe, firstform, firstformdiag, secondform, thirdform, shape, principals, principalaxes, curvatures, meancurvature, gaussintrinsic, gaussextrinsic, gausssign.

The Cartan package is intended to standardize the composition of various methods and functors applied to specialized categories transformed with a unified representation over a product topology, especially having fibers of the Grassmann algebra. Initial topologies include ProductSpace types and in general the ImmersedTopology.

Positions{P, G} where {P<:Chain, G} (alias for AbstractArray{<:Coordinate{P, G}, 1} where {P<:Chain, G})
Interval{P, G} where {P<:AbstractReal, G} (alias for AbstractArray{<:Coordinate{P, G}, 1} where {P<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, G})
IntervalRange{P, G, PA, GA} where {P<:Real, G, PA<:AbstractRange, GA} (alias for GridBundle{1, Coordinate{P, G}, <:PointArray{P, G, 1, PA, GA}} where {P<:Real, G, PA<:AbstractRange, GA})
Rectangle (alias for ProductSpace{V, T, 2, 2} where {V, T})
Hyperrectangle (alias for ProductSpace{V, T, 3, 3} where {V, T})
RealRegion{V, T} where {V, T<:Real} (alias for ProductSpace{V, T, N, N, S} where {V, T<:Real, N, S<:AbstractArray{T, 1}})
RealSpace{N} where N (alias for AbstractArray{<:Coordinate{P, G}, N} where {N, P<:(Chain{V, 1, <:Real} where V), G})
AlignedRegion{N} where N (alias for GridBundle{N, Coordinate{P, G}, PointArray{P, G, N, PA, GA}} where {N, P<:Chain, G<:InducedMetric, PA<:(ProductSpace{V, <:Real, N, N, <:AbstractRange} where V), GA<:Global})
AlignedSpace{N} where N (alias for GridBundle{N, Coordinate{P, G}, PointArray{P, G, N, PA, GA}} where {N, P<:Chain, G<:InducedMetric, PA<:(ProductSpace{V, <:Real, N, N, <:AbstractRange} where V), GA})
FrameBundle{Coordinate{B,F},N} where {B,F,N}
GridBundle{N,C,PA<:FiberBundle{C,N},TA<:ImmersedTopology} <: FrameBundle{C,N}
SimplexBundle{N,C,PA<:FiberBundle{C,1},TA<:ImmersedTopology} <: FrameBundle{C,1}
FaceBundle{N,C,PA<:FiberBundle{C,1},TA<:ImmersedTopology} <: FrameBundle{C,1}
FiberProductBundle{P,N,SA<:AbstractArray,PA<:AbstractArray} <: FrameBundle{Coordinate{P,InducedMetric},N}
HomotopyBundle{P,N,PA<:AbstractArray{F,N} where F,FA<:AbstractArray,TA<:ImmersedTopology} <: FrameBundle{Coordinate{P,InducedMetric},N}

Visualizing TensorField reperesentations can be standardized in combination with Makie.jl or UnicodePlots.jl.

Due to the versatility of the TensorField type instances, it's possible to disambiguate them into these type alias specifications with associated methods:

ElementMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:ElementBundle, A})
SimplexMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:SimplexBundle, A})
FaceMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:FaceBundle, A})
IntervalMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:(AbstractArray{<:Coordinate{P, G}, 1} where {P<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, G}), A})
RectangleMap (alias for TensorField{B, F, 2, P, A} where {B, F, P<:(AbstractMatrix{<:Coordinate{P, G}} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
HyperrectangleMap (alias for TensorField{B, F, 3, P, A} where {B, F, P<:(AbstractArray{<:Coordinate{P, G}, 3} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
ParametricMap (alias for TensorField{B, F, N, P, A} where {B, F, N, P<:(AbstractArray{<:Coordinate{P, G}, N} where {N, P<:(Chain{V, 1, <:Real} where V), G}), A})
Variation (alias for TensorField{B, F, N, P, A} where {B, F<:TensorField, N, P, A})
RealFunction (alias for TensorField{B, F, 1, P, A} where {B, F<:AbstractReal, PA<:(AbstractVector{<:AbstractReal}), A})
PlaneCurve (alias for TensorField{B, F, N, P, A} where {B, F, N, P<:(AbstractArray{<:Coordinate{P, G}, N} where {N, P<:(Chain{V, 1, <:Real} where V), G}), A})
SpaceCurve (alias for TensorField{B, F, 1, P, A} where {B, F<:(Chain{V, G, Q, 3} where {V, G, Q}), P<:(AbstractVector{<:Coordinate{P, G}} where {P<:AbstractReal, G}), A})
AbstractCurve (alias for TensorField{B, F, 1, P, A} where {B, F<:Chain, P<:(AbstractVector{<:Coordinate{P, G}} where {P<:AbstractReal, G}), A})
SurfaceGrid (alias for TensorField{B, F, 2, P, A} where {B, F<:AbstractReal, P<:(AbstractMatrix{<:Coordinate{P, G}} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
VolumeGrid (alias for TensorField{B, F, 3, P, A} where {B, F<:AbstractReal, P<:(AbstractArray{<:Coordinate{P, G}, 3} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
ScalarGrid (alias for TensorField{B, F, N, P, A} where {B, F<:AbstractReal, N, P<:(AbstractArray{<:Coordinate{P, G}, N} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
DiagonalField (alias for TensorField{B, F, N, P, A} where {B, F<:DiagonalOperator, N, P, A})
EndomorphismField (alias for TensorField{B, F, N, P, A} where {B, F<:(TensorOperator{V, V, T} where {V T<:(TensorAlgebra{V, <:TensorAlgebra{V}})}), N, P, A})
OutermorphismField (alias for TensorField{B, F, N, P, A} where {B, F<:Outermorphism, N, P, A})
CliffordField (alias for TensorField{B, F, N, P, A} where {B, F<:Multivector, N, P, A})
QuaternionField (alias for TensorField{B, F, N, P, A} where {B, F<:(Quaternion), N, P, A})
ComplexMap (alias for TensorField{B, F, N, P, A} where {B, F<:(Union{Complex{T}, Single{V, G, B, Complex{T}} where {V, G, B}, Chain{V, G, Complex{T}, 1} where {V, G}, Couple{V, B, T} where {V, B}, Phasor{V, B, T} where {V, B}} where T<:Real), N, P, A})
PhasorField (alias for TensorField{B, F, N, P, A} where {B, F<:Phasor, N, P, A})
SpinorField (alias for TensorField{B, F, N, P, A} where {B, F<:AbstractSpinor, N, P, A})
GradedField{G} where G (alias for TensorField{B, F, N, P, A} where {G, B, F<:(Chain{V, G} where V), N, P, A})
ScalarField (alias for TensorField{B, F, N, P, A} where {B, F<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, N, P, A})
VectorField (alias for TensorField{B, F, N, P, A} where {B, F<:(Chain{V, 1} where V), N, P, A})
BivectorField (alias for TensorField{B, F, N, P, A} where {B, F<:(Chain{V, 2} where V), N, P, A})
TrivectorField (alias for TensorField{B, F, N, P, A} where {B, F<:(Chain{V, 3} where V),N, P, A})

In the Cartan package, a technique is employed where a TensorField is constructed from an interval, product manifold, or topology, to generate an algebra of sections which can be used to compose parametric maps on manifolds. Constructing a TensorField can be accomplished in various ways, there are explicit techniques to construct a TensorField as well as implicit methods. Additional packages such as Adapode build on the TensorField concept by generating them from differential equations. Many of these methods can automatically generalize to higher dimensional manifolds and are compatible with discrete differential geometry.

References

• Michael Reed, Differential geometric algebra with Leibniz and Grassmann (2019)
• Michael Reed, Foundatons of differential geometric algebra (2021)
• Michael Reed, Multilinear Lie bracket recursion formula (2024)
• Michael Reed, Differential geometric algebra: compute using Grassmann.jl and Cartan.jl (2025)
• Emil Artin, Geometric Algebra (1957)
• John Browne, Grassmann Algebra, Volume 1: Foundations (2011)
• C. Doran, D. Hestenes, F. Sommen, and N. Van Acker, Lie groups as spin groups, J. Math Phys. (1993)
• David Hestenes, Tutorial on geometric calculus. Advances in Applied Clifford Algebra (2013)
• Vladimir and Tijana Ivancevic, Undergraduate lecture notes in DeRahm-Hodge theory. arXiv (2011)