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.

SymbolDescriptionAlso usedComment
$\tau_p$action map by group element $p$$\mathrm{L}_p, \mathrm{R}_peither left or right \operatorname{Ad}_p(X)adjoint action of element p of a Lie group on the element X of the corresponding Lie algebra ×Cartesian product of two manifoldssee ProductManifold ^{\wedge}(n-ary) Cartesian power of a manifoldsee PowerManifold ⋅^\mathrm{H}conjugate/Hermitian transpose acoordinates of a point in a chartsee get_parameters \frac{\mathrm{D}}{\mathrm{d}t}covariant derivative of a vector field X(t) T^*_p \mathcal Mthe cotangent space at p ξa cotangent vector from T^*_p \mathcal M$$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$$\exp_p(X) Fa fibersee Fiber \mathbb Fa field, usually \mathbb F \in \{\mathbb R,\mathbb C, \mathbb H\}, i.e. the real, complex, and quaternion numbers, respectively.field a manifold or a basis is based on \gammaa geodesic\gamma_{p;q}, \gamma_{p,X}connecting two points p,q or starting in p with velocity X. \operatorname{grad} f(p)(Riemannian) gradient of function f \colon \mathcal{M} → ℝ at p \in \mathcal{M} \nabla f(p)(Euclidean) gradient of function f \colon \mathcal{M} → ℝ at p \in \mathcal{M} but thought of as evaluated in the embeddingG \circa group operation ⋅^\mathrm{H}Hermitian or conjugate transposed for both complex or quaternion matrices \operatorname{Hess} f(p)(Riemannian) Hessian of function f \colon T_p\mathcal{M} → T_p\mathcal M (i.e. the 1-1-tensor form) at p \in \mathcal{M} \nabla^2 f(p)(Euclidean) Hessian of function f in the embeddingH eidentity element of a group I_kidentity matrix of size k×k kindicesi,j \langle⋅,⋅\rangleinner product (in T_p \mathcal M)\langle⋅,⋅\rangle_p, g_p(⋅,⋅) \operatorname{retr}^{-1}_pqan inverse retraction \mathfrak ga Lie algebra \mathcal{G}a (Lie) group \log_p qlogarithmic map at p \in \mathcal M of a point q \in \mathcal M$$\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$$W \operatorname{Exp}the matrix exponential \operatorname{Log}the matrix logarithm \mathcal P_{q\gets p}Xparallel transportof the vector X from T_p\mathcal M to T_q\mathcal M \mathcal P_{p,Y}Xparallel transport in direction Yof the vector X from T_p\mathcal M to T_q\mathcal M, q = \exp_pY \mathcal P_{t_1\gets t_0}^cXparallel transport along the curve c$$\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$$p_1, p_2, \ldots,qfor 3 points one might use x,y,z \operatorname{retr}_pXa retraction \kappa_p(X, Y)sectional curvature ξa set of tangent vectors\{X_1,\ldots,X_n\} J_{2n} \in ℝ^{2n×2n}the SymplecticElement T_p \mathcal Mthe tangent space at p Xa tangent vector from T_p \mathcal M$$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 $i$th 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 transportof 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.