# 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$$\mathrm{L}_p, \mathrm{R}_peither left or right \timesCartesian product of two manifoldssee ProductManifold ^{\wedge}(n-ary) Cartesian power of a manifoldsee PowerManifold T^*_p \mathcal Mthe cotangent space at p \xia cotangent vector from T^*_p \mathcal M$$\xi_1, \xi_2,\ldots,\eta,\zeta$sometimes written with base point $\xi_p$.
$\mathrm{d}\phi_p$differential of a map $\phi: \mathcal M \to \mathcal N$ at a point $p$$(\mathrm{d}\phi)_p, (\phi_*)_ppushes tangent vectors in T_p \mathcal M forward to T_{\phi(p)} \mathcal N ndimension (of a manifold)n_1,n_2,\ldots,m, \dim(\mathcal M)for the real dimension sometimes also \dim_{\mathbb R}(\mathcal M) d(\cdot,\cdot)(Riemannian) distanced_{\mathcal M}(\cdot,\cdot) Fa fiber \mathbb Fa fieldfield a manifold is based on, usually \mathbb F \in \{\mathbb R,\mathbb C\} \gammaa geodesic\gamma_{p;q}, \gamma_{p,X}connecting two points p,q or starting in p with velocity X. \circa group operation \cdot^\mathrm{H}Hermitian or conjugate transposed eidentity element of a group I_kidentity matrix of size k\times k kindicesi,j \langle\cdot,\cdot\rangleinner product (in T_p \mathcal M)\langle\cdot,\cdot\rangle_p, g_p(\cdot,\cdot) \mathfrak ga Lie algebra \mathcal{G}a (Lie) group \mathcal Ma manifold\mathcal M_1, \mathcal M_2,\ldots,\mathcal N \operatorname{Exp}the matrix exponential \operatorname{Log}the matrix logarithm \mathcal P_{q\gets p}Xparallel transportof the vector X from T_p\mathcal M to T_q\mathcal M pa point on \mathcal M$$p_1, p_2, \ldots,q$for 3 points one might use $x,y,z$
$\Xi$a set of tangent vectors$\{X_1,\ldots,X_n\}$
$T_p \mathcal M$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)
$\cdot^\mathrm{T}$transposed
$B$a vector bundle
$0_k$the $k\times k$ zero matrix.