Data.Manifold.FibreBundle

Documentation

pattern TangentBundle :: m -> Needle m -> FibreBundle m (Needle m)

pattern (:@.) :: f -> m -> FibreBundle m f infixr 5

Provided for convenience. Flipped synonym of FibreBundle, restricted to manifolds without boundary (so the type of the whole can be inferred from its interior).

tangentAt :: (AdditiveGroup (Needle m), m ~ Interior m) => m -> TangentBundle m

A zero vector in the fibre bundle at the given position. Intended to be used with tangent-modifying lenses such as delta.

data TransportOnNeedleWitness k m f where

Constructors

TransportOnNeedle :: ParallelTransporting (LinearFunction (Scalar (Needle m))) (Needle m) (Needle f) => TransportOnNeedleWitness k m f

data ForgetTransportProperties k m f where

Constructors

ForgetTransportProperties :: ParallelTransporting (->) m f => ForgetTransportProperties k m f

class (PseudoAffine m, m ~ Interior m, Category k, Object k f) => ParallelTransporting k m f where

Minimal complete definition

parallelTransport

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k m f

transportOnNeedleWitness :: ParallelTransporting (LinearFunction (Scalar (Needle m))) (Needle m) (Needle f) => TransportOnNeedleWitness k m f

forgetTransportProperties :: ForgetTransportProperties k m f

forgetTransportProperties :: ParallelTransporting (->) m f => ForgetTransportProperties k m f

parallelTransport :: m -> Needle m -> k f f

translateAndInvblyParTransport :: m -> Needle m -> (m, (k f f, k f f))

Instances

(EnhancedCat k (LinearMap ℝ), Object k ℝ²) => ParallelTransporting k S² ℝ²

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k S² ℝ²

forgetTransportProperties :: ForgetTransportProperties k S² ℝ²

parallelTransport :: S² -> Needle S² -> k ℝ² ℝ²

translateAndInvblyParTransport :: S² -> Needle S² -> (S², (k ℝ² ℝ², k ℝ² ℝ²))

(Category k, Object k ℝ) => ParallelTransporting k S¹ ℝ

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k S¹ ℝ

forgetTransportProperties :: ForgetTransportProperties k S¹ ℝ

parallelTransport :: S¹ -> Needle S¹ -> k ℝ ℝ

translateAndInvblyParTransport :: S¹ -> Needle S¹ -> (S¹, (k ℝ ℝ, k ℝ ℝ))

(Category k, Object k ℝ⁴) => ParallelTransporting k ℝ⁴ ℝ⁴

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ⁴ ℝ⁴

forgetTransportProperties :: ForgetTransportProperties k ℝ⁴ ℝ⁴

parallelTransport :: ℝ⁴ -> Needle ℝ⁴ -> k ℝ⁴ ℝ⁴

translateAndInvblyParTransport :: ℝ⁴ -> Needle ℝ⁴ -> (ℝ⁴, (k ℝ⁴ ℝ⁴, k ℝ⁴ ℝ⁴))

(Category k, Object k ℝ³) => ParallelTransporting k ℝ³ ℝ³

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ³ ℝ³

forgetTransportProperties :: ForgetTransportProperties k ℝ³ ℝ³

parallelTransport :: ℝ³ -> Needle ℝ³ -> k ℝ³ ℝ³

translateAndInvblyParTransport :: ℝ³ -> Needle ℝ³ -> (ℝ³, (k ℝ³ ℝ³, k ℝ³ ℝ³))

(Category k, Object k ℝ²) => ParallelTransporting k ℝ² ℝ²

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ² ℝ²

forgetTransportProperties :: ForgetTransportProperties k ℝ² ℝ²

parallelTransport :: ℝ² -> Needle ℝ² -> k ℝ² ℝ²

translateAndInvblyParTransport :: ℝ² -> Needle ℝ² -> (ℝ², (k ℝ² ℝ², k ℝ² ℝ²))

(Category k, Object k ℝ) => ParallelTransporting k ℝ ℝ

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ ℝ

forgetTransportProperties :: ForgetTransportProperties k ℝ ℝ

parallelTransport :: ℝ -> Needle ℝ -> k ℝ ℝ

translateAndInvblyParTransport :: ℝ -> Needle ℝ -> (ℝ, (k ℝ ℝ, k ℝ ℝ))

(ParallelTransporting k a f, ParallelTransporting k a g, ParallelTransporting (LinearFunction s) (Needle a) (Needle f, Needle g), PseudoAffine f, PseudoAffine g, Morphism k, ObjectPair k f g) => ParallelTransporting k a (f, g)

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k a (f, g)

forgetTransportProperties :: ForgetTransportProperties k a (f, g)

parallelTransport :: a -> Needle a -> k (f, g) (f, g)

translateAndInvblyParTransport :: a -> Needle a -> (a, (k (f, g) (f, g), k (f, g) (f, g)))

(ParallelTransporting k a fa, ParallelTransporting k b fb, PseudoAffine fa, PseudoAffine fb, (~) * (Scalar (Needle a)) s, (~) * (Scalar (Needle b)) s, (~) * (Scalar (Needle fa)) s, (~) * (Scalar (Needle fb)) s, Num' s, Morphism k, ObjectPair k fa fb) => ParallelTransporting k (a, b) (fa, fb)

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k (a, b) (fa, fb)

forgetTransportProperties :: ForgetTransportProperties k (a, b) (fa, fb)

parallelTransport :: (a, b) -> Needle (a, b) -> k (fa, fb) (fa, fb)

translateAndInvblyParTransport :: (a, b) -> Needle (a, b) -> ((a, b), (k (fa, fb) (fa, fb), k (fa, fb) (fa, fb)))

(PseudoAffine m, (~) * m (Interior m), (~) * s (Scalar (Needle m)), Num' s) => ParallelTransporting (Discrete *) m (ZeroDim s)

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness (Discrete *) m (ZeroDim s)

forgetTransportProperties :: ForgetTransportProperties (Discrete *) m (ZeroDim s)

parallelTransport :: m -> Needle m -> Discrete * (ZeroDim s) (ZeroDim s)

translateAndInvblyParTransport :: m -> Needle m -> (m, (Discrete * (ZeroDim s) (ZeroDim s), Discrete * (ZeroDim s) (ZeroDim s)))

(PseudoAffine m, (~) * m (Interior m), (~) * s (Scalar (Needle m)), Num' s) => ParallelTransporting (LinearFunction s) m (ZeroDim s)

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness (LinearFunction s) m (ZeroDim s)

forgetTransportProperties :: ForgetTransportProperties (LinearFunction s) m (ZeroDim s)

parallelTransport :: m -> Needle m -> LinearFunction s (ZeroDim s) (ZeroDim s)

translateAndInvblyParTransport :: m -> Needle m -> (m, (LinearFunction s (ZeroDim s) (ZeroDim s), LinearFunction s (ZeroDim s) (ZeroDim s)))

(PseudoAffine m, (~) * m (Interior m), (~) * s (Scalar (Needle m)), Num' s) => ParallelTransporting ((->) LiftedRep LiftedRep) m (ZeroDim s)

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness (LiftedRep -> LiftedRep) m (ZeroDim s)

forgetTransportProperties :: ForgetTransportProperties (LiftedRep -> LiftedRep) m (ZeroDim s)

parallelTransport :: m -> Needle m -> (LiftedRep -> LiftedRep) (ZeroDim s) (ZeroDim s)

translateAndInvblyParTransport :: m -> Needle m -> (m, ((LiftedRep -> LiftedRep) (ZeroDim s) (ZeroDim s), (LiftedRep -> LiftedRep) (ZeroDim s) (ZeroDim s)))

transformEmbeddedTangents :: forall x f v. (NaturallyEmbedded (FibreBundle x f) (FibreBundle v v), v ~ Interior v) => (v -> v) -> FibreBundle x f -> FibreBundle x f

ex -> ey, ey -> ez, ez -> ex

Orphan instances

(AdditiveGroup f, (~) * x (Interior x)) => NaturallyEmbedded x (FibreBundle x f)

Methods

embed :: x -> FibreBundle x f

coEmbed :: FibreBundle x f -> x

(ParallelTransporting (LinearFunction (Scalar f)) m f, AdditiveGroup m, VectorSpace f) => AdditiveGroup (FibreBundle m f)

Methods

zeroV :: FibreBundle m f

(^+^) :: FibreBundle m f -> FibreBundle m f -> FibreBundle m f

negateV :: FibreBundle m f -> FibreBundle m f

(^-^) :: FibreBundle m f -> FibreBundle m f -> FibreBundle m f

(ParallelTransporting ((->) LiftedRep LiftedRep) m (Interior f), Semimanifold f, ParallelTransporting (LinearFunction s) (Needle m) (Needle f), (~) * s (Scalar (Needle m))) => Semimanifold (FibreBundle m f)

Associated Types

type Needle (FibreBundle m f) :: *

type Interior (FibreBundle m f) :: *

Methods

(.+~^) :: Interior (FibreBundle m f) -> Needle (FibreBundle m f) -> FibreBundle m f

fromInterior :: Interior (FibreBundle m f) -> FibreBundle m f

toInterior :: FibreBundle m f -> Maybe (Interior (FibreBundle m f))

translateP :: Tagged * (FibreBundle m f) (Interior (FibreBundle m f) -> Needle (FibreBundle m f) -> Interior (FibreBundle m f))

(.-~^) :: Interior (FibreBundle m f) -> Needle (FibreBundle m f) -> FibreBundle m f

semimanifoldWitness :: SemimanifoldWitness (FibreBundle m f)

(ParallelTransporting ((->) LiftedRep LiftedRep) m f, ParallelTransporting ((->) LiftedRep LiftedRep) m (Interior f), PseudoAffine f, ParallelTransporting (LinearFunction s) (Needle m) (Needle f), (~) * s (Scalar (Needle m))) => PseudoAffine (FibreBundle m f)

Methods

(.-~.) :: FibreBundle m f -> FibreBundle m f -> Maybe (Needle (FibreBundle m f))

(.-~!) :: FibreBundle m f -> FibreBundle m f -> Needle (FibreBundle m f)

pseudoAffineWitness :: PseudoAffineWitness (FibreBundle m f)

Rotatable (FibreBundle S² ℝ²)

Associated Types

type AxisSpace (FibreBundle S² ℝ²) :: *

Methods

rotateAbout :: AxisSpace (FibreBundle S² ℝ²) -> S¹ -> FibreBundle S² ℝ² -> FibreBundle S² ℝ²

(AdditiveGroup y, AdditiveGroup g) => NaturallyEmbedded (FibreBundle x f) (FibreBundle (x, y) (f, g))

Methods

embed :: FibreBundle x f -> FibreBundle (x, y) (f, g)

coEmbed :: FibreBundle (x, y) (f, g) -> FibreBundle x f

(NaturallyEmbedded m v, VectorSpace f) => NaturallyEmbedded (FibreBundle m ℝ⁰) (FibreBundle v f)

Methods

embed :: FibreBundle m ℝ⁰ -> FibreBundle v f

coEmbed :: FibreBundle v f -> FibreBundle m ℝ⁰

NaturallyEmbedded (FibreBundle S¹ ℝ) (FibreBundle ℝ² ℝ²)

Methods

embed :: FibreBundle S¹ ℝ -> FibreBundle ℝ² ℝ²

coEmbed :: FibreBundle ℝ² ℝ² -> FibreBundle S¹ ℝ

NaturallyEmbedded (FibreBundle S² ℝ²) (FibreBundle ℝ³ ℝ³)

Methods

embed :: FibreBundle S² ℝ² -> FibreBundle ℝ³ ℝ³

coEmbed :: FibreBundle ℝ³ ℝ³ -> FibreBundle S² ℝ²

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ v) (FibreBundle ℝ w)

Methods

embed :: FibreBundle ℝ v -> FibreBundle ℝ w

coEmbed :: FibreBundle ℝ w -> FibreBundle ℝ v

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ⁴ v) (FibreBundle ℝ⁴ w)

Methods

embed :: FibreBundle ℝ⁴ v -> FibreBundle ℝ⁴ w

coEmbed :: FibreBundle ℝ⁴ w -> FibreBundle ℝ⁴ v

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ³ v) (FibreBundle ℝ³ w)

Methods

embed :: FibreBundle ℝ³ v -> FibreBundle ℝ³ w

coEmbed :: FibreBundle ℝ³ w -> FibreBundle ℝ³ v

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ² v) (FibreBundle ℝ² w)

Methods

embed :: FibreBundle ℝ² v -> FibreBundle ℝ² w

coEmbed :: FibreBundle ℝ² w -> FibreBundle ℝ² v