notation.md

Notation overview

Since manifolds include a reasonable amount of elements and functions, the following list tries to keep an overview of used notation throughout Manifolds.jl. The order is alphabetical by name. They might be used in a plain form within the code or when referring to that code. This is for example the case with the calligraphic symbols.

Within the documented functions, the utf8 symbols are used whenever possible, as long as that renders correctly in \TeX within this documentation.

Symbol	Description	Also used	Comment
\tau_p	action map by group element p	\mathrm{L}_p, \mathrm{R}_p	either left or right
\operatorname{Ad}_p(X)	adjoint action of element p of a Lie group on the element X of the corresponding Lie algebra
×	Cartesian product of two manifolds		see [ProductManifold](@extref ManifoldsBase.ProductManifold)
^{\wedge}	(n-ary) Cartesian power of a manifold		see [PowerManifold](@extref ManifoldsBase.PowerManifold)
⋅^\mathrm{H}	conjugate/Hermitian transpose
a	coordinates of a point in a chart		see get_parameters
\frac{\mathrm{D}}{\mathrm{d}t}	covariant derivative of a vector field X(t)
T^*_p \mathcal M	the cotangent space at p
ξ	a cotangent vector from T^*_p \mathcal M	ξ_1, ξ_2,… ,η,\zeta	sometimes written with base point ξ_p.
\mathrm{d}\phi_p(q)	Differential of a map \phi: \mathcal M → \mathcal N with respect to p at a point q. For functions of multiple variables, for example \phi(p, p_1) where p \in \mathcal M and p_1 \in \mathcal M_1, variable p is explicitly stated to specify with respect to which argument the differential is calculated.	\mathrm{d}\phi_q, (\mathrm{d}\phi)_q, (\phi_*)_q, D_p\phi(q)	pushes tangent vectors X \in T_q \mathcal M forward to \mathrm{d}\phi_p(q)[X] \in T_{\phi(q)} \mathcal N
n	dimension (of a manifold)	n_1,n_2,\ldots,m, \dim(\mathcal M)	for the real dimension sometimes also \dim_{\mathbb R}(\mathcal M)
d(⋅,⋅)	(Riemannian) distance	d_{\mathcal M}(⋅,⋅)
\exp_p X	exponential map at p \in \mathcal M of a vector X \in T_p \mathcal M	\exp_p(X)
F	a fiber		see [Fiber](@extref ManifoldsBase.Fiber)
\mathbb F	a field, usually \mathbb F \in \{\mathbb R,\mathbb C, \mathbb H\}, i.e. the real, complex, and quaternion numbers, respectively.		field a manifold or a basis is based on
\gamma	a geodesic	\gamma_{p;q}, \gamma_{p,X}	connecting two points p,q or starting in p with velocity X.
\operatorname{grad} f(p)	(Riemannian) gradient of function f \colon \mathcal{M} → ℝ at p \in \mathcal{M}
\nabla f(p)	(Euclidean) gradient of function f \colon \mathcal{M} → ℝ at p \in \mathcal{M} but thought of as evaluated in the embedding	G
\circ	a group operation
⋅^\mathrm{H}	Hermitian or conjugate transposed for both complex or quaternion matrices
\operatorname{Hess} f(p)	(Riemannian) Hessian of function f \colon T_p\mathcal{M} → T_p\mathcal M (i.e. the 1-1-tensor form) at p \in \mathcal{M}
\nabla^2 f(p)	(Euclidean) Hessian of function f in the embedding	H
e	identity element of a group
I_k	identity matrix of size k×k
k	indices	i,j
\langle⋅,⋅\rangle	inner product (in T_p \mathcal M)	\langle⋅,⋅\rangle_p, g_p(⋅,⋅)
\operatorname{retr}^{-1}_pq	an inverse retraction
\mathfrak g	a Lie algebra
\mathcal{G}	a (Lie) group
\log_p q	logarithmic map at p \in \mathcal M of a point q \in \mathcal M	\log_p(q)
\mathcal M	a manifold	\mathcal M_1, \mathcal M_2,\ldots,\mathcal N
N_p \mathcal M	the normal space of the tangent space T_p \mathcal M in some embedding \mathcal E that should be clear from context
V	a normal vector from N_p \mathcal M	W
\operatorname{Exp}	the matrix exponential
\operatorname{Log}	the matrix logarithm
\mathcal P_{q\gets p}X	parallel transport		of the vector X from T_p\mathcal M to T_q\mathcal M
\mathcal P_{p,Y}X	parallel transport in direction Y		of the vector X from T_p\mathcal M to T_q\mathcal M, q = \exp_pY
\mathcal P_{t_1\gets t_0}^cX	parallel transport along the curve c	\mathcal P^cX=\mathcal P_{1\gets 0}^cX	of the vector X from p=c(0) to c(1)
p	a point on \mathcal M	p_1, p_2, \ldots,q	for 3 points one might use x,y,z
\operatorname{retr}_pX	a retraction
\kappa_p(X, Y)	sectional curvature
ξ	a set of tangent vectors	\{X_1,\ldots,X_n\}
J_{2n} \in ℝ^{2n×2n}	the SymplecticElement
T_p \mathcal M	the tangent space at p
X	a tangent vector from T_p \mathcal M	X_1,X_2,\ldots,Y,Z	sometimes written with base point X_p
\operatorname{tr}	trace (of a matrix)
⋅^\mathrm{T}	transposed
e_i \in \mathbb R^n	the ith unit vector	e_i^n	the space dimension (n) is omitted, when clear from context
B	a vector bundle
\mathcal T_{q\gets p}X	vector transport		of the vector X from T_p\mathcal M to T_q\mathcal M
\mathcal T_{p,Y}X	vector transport in direction Y		of the vector X from T_p\mathcal M to T_q\mathcal M, where q is determined by Y, for example using the exponential map or some retraction.
\operatorname{Vol}(\mathcal M)	volume of manifold \mathcal M
\theta_p(X)	volume density for vector X tangent at point p
\mathcal W	the Weingarten map \mathcal W: T_p\mathcal M × N_p\mathcal M → T_p\mathcal M	\mathcal W_p	the second notation to emphasize the dependency of the point p\in\mathcal M
0_k	the k×k zero matrix.

Comparison with notation commonly used in robotics

In robotics, a different notation is commonly used. The table below shows a quick guide how to translate between them for people coming from robotics background. We use SolaDerayAtchuthan:2021 as the primary robotics source.

Robotics concept	Manifolds.jl notation
p \circ q	compose(G, p, q)
p^{-1}	inv(G, p)
\mathcal{E}	Identity(G) or identity_element(G)
group action p\cdot p_m	apply(A, p, p_m)
Lie group exponential \exp\colon \mathfrak{g} \to \mathcal{G}, \exp(X)=p	exp_lie(G, p)
Lie group logarithm \log\colon \mathcal{G} \to \mathfrak{g}, \log(p)=X	log_lie(G, X)
n-D vector	TranslationGroup(n); its action is TranslationAction(Euclidean(n), TranslationGroup(n))
circle S^1	CircleGroup(); its action is ComplexPlanarRotation
rotation \mathrm{SO}(n)	SpecialOrthogonal(n); its action is RotationAction(Euclidean(n), SpecialOrthogonal(n))
rigid motion \mathrm{SE}(n)	SpecialEuclidean(n); its action is RotationTranslationAction(Euclidean(n), SpecialEuclidean(n))
unit quaternions S^3	UnitaryMatrices(1, H); note that 3-sphere and the group of rotations (with its bi-invariant metric) are homeomorphic but not isomorphic
size (as in Table I)	related to representation_size(G)
dim (as in Table I)	manifold_dimension(G)
Lie algebra element with coordinates \tau^{\wedge}	hat(G, Identity(G), tau)
coordinates of an element of Lie algebra X^{\vee}	vee(G, Identity(G), X)
capital exponential map \operatorname{Exp}	exp_lie(G, hat(G, Identity(G), tau))
capital logarithmic map \operatorname{Log}	vee(G, Identity(G), log_lie(G, p))
right-\oplus, p \oplus \tau	compose(G, exp_lie(G, hat(G, Identity(G), tau)))
right-\ominus, p \ominus q	vee(G, Identity(G), log_lie(G, compose(G, inv(G, q), p)))
left-\oplus, \tau \oplus p	compose(G, exp_lie(G, hat(G, Identity(G), tau)), p)
left-\ominus, p \ominus q	vee(G, Identity(G), log_lie(G, compose(G, p, inv(G, q))))
adjoint \mathrm{Ad}_{p}(\tau^{\wedge})	adjoint_action(G, p, hat(G, Identity(G), tau))
adjoint matrix \mathrm{Ad}_{p}	adjoint_matrix(G, p)
Jacobian of group inversion and composition	these can be easily constructed from the adjoint matrix
left and right Jacobians of a function	In JuliaManifolds there is always one preferred way to store tangent vectors specified by each manifold, and so we follow the standard mathematical convention of having one Jacobian which follows the selected tangent vector storage convention. See for example jacobian_exp_argument, jacobian_exp_basepoint, jacobian_log_argument, jacobian_log_basepoint from ManifoldDiff.jl.
left and right Jacobians (of a group) \mathbf{J}_l, \mathbf{J}_r	jacobian_exp_argument for exponential coordinates. For other coordinate systems no replacement is available yet.
Jacobians of group actions	not available yet

Be also careful that the meaning of \mathbf{x} is inconsistent in Table I from SolaDerayAtchuthan:2021. It's a complex number for circle, quaternion for quaternion rotation and column vectors for other rows.