# Defined notations

The following are symbols and conventions used for relations and structures defined in this book. Most are reflective of notation commonly used by other authors, but some are noted as particular to this book.

 $${\cong}$$ Isomorphism (between algebraic objects), homeomorphism (between topological spaces), diffeomorphism (between differential manifolds), or isometry (between Riemannian manifolds) $${\simeq}$$ Homotopy equivalency (between topological spaces) $${\textbf{1}}$$ Identity element in a monoid $${\mathbf{0}}$$ The zero element in a ring $${\textrm{Ker},\textrm{Im}}$$ The kernel and image of a mapping $${|G|;|g|}$$ Order of a group $${G}$$ or element $${g}$$ $${\textrm{Aut}(X)}$$ The group of all automorphisms of $${X}$$ (group, space, manifold, etc) $${\textrm{Inn}(G)}$$ The group of all inner automorphisms of a group $${G}$$ $${N\triangleleft G}$$ $${N}$$ is a normal subgroup of $${G}$$ $${G=N\rtimes H}$$ $${G}$$ is the semidirect product of a normal $${N}$$ and $${H}$$ $${|G:H|}$$ Index of a group $${G}$$ over a subgroup $${H}$$ $${G^{e}}$$ Identity component of a topological group $${V^{\perp}}$$ Orthogonal complement of a vector space $${V}$$ $${V^{*}}$$ The dual space of $${V}$$ $${\delta^{\mu}{}_{\nu},\delta_{\mu\nu}}$$ The Kronecker delta $${\equiv}$$ 1 if $${\mu=\nu}$$, 0 otherwise $${\eta^{\mu}{}_{\nu},\eta{}_{\mu\nu}}$$ For a pseudo inner product of signature $${\left(r,s\right)}$$, $${\pm1}$$ if $${\mu=\nu}$$ (with $${r}$$ positive values), $${0}$$ otherwise $${T^{k}V}$$ The $${k^{\textrm{th}}}$$ tensor power of $${V}$$ $${\Lambda^{k}V}$$ The $${k^{\textrm{th}}}$$ exterior power of $${V}$$ $${M^n}$$ Manifold of dimension $${n}$$ $${\Lambda^{k}M}$$ Differential $${k}$$-forms defined on a manifold $${M}$$ $${T_{x}M,TM}$$ Tangent space at $${x}$$, tangent bundle on $${M}$$ $${FM}$$ Frame bundle on $${M}$$ $${\mathrm{Diff}(M)}$$ The Lie group of diffeomorphisms of a manifold $${\mathrm{vect}(M)}$$ The Lie algebra of vector fields on a manifold $${\left\langle v,w\right\rangle }$$ Inner product of two vectors $${\left\Vert v\right\Vert }$$ Norm $${\left[u,v\right]}$$ Lie bracket $${*A}$$ Hodge star of $${A\in\Lambda^{k}V}$$ $${\Omega}$$ Unit $${n}$$-vector (non-standard) $${\widetilde{A}}$$ Reverse of a Clifford algebra element (in geometric algebra) $${V\times W}$$ Direct product of two vector spaces $${V\oplus W}$$ Direct sum $${V*W}$$ Free product $${V\otimes W}$$ Tensor product $${V\wedge W}$$ Exterior product $${\check{\Theta}[\wedge]\check{\Psi}}$$ Exterior product of Lie algebra valued forms using the Lie commutator (non-standard) $${\check{\Theta}\wedge\check{\Psi}}$$ Exterior product of Lie algebra valued forms using the multiplication of the related associative algebra $${X\times Y}$$ Product of two topological spaces $${X\vee Y}$$ Wedge sum of two spaces $${X*Y}$$ Join of two spaces $${\mathbb{R}\mathrm{P}^{n}}$$ Real projective $${n}$$-space $${H^{n}}$$ Real hyperbolic $${n}$$-space