$\left[D\right]$	$\left[D\right]:=\{1,\dots ,D\}$
$\langle ·,·\rangle$	inner product
$M_Q(u,v)={\parallel u-v\parallel }_Q$	Mahalanobis distance $M_Q(u,v)=\sqrt{{\sum }_{i,j}(u^i-v^i)(u^j-v^j)Q_{ij}}$, $Q\succ 0$
$D(\theta :{\theta }^{\prime })$	parameter divergence
$D[p\left(x\right):p^{\prime }\left(x\right)]$	statistical divergence
D, $D^{*}$	Divergence and dual (reverse) divergence
Csiszár divergence $I_f$	$I_f(\theta :{\theta }^{\prime }):={\sum }_{i=1}^D{\theta }_{i}f\left(\frac{{\theta }_i^{\prime }}{{\theta }_i}\right)$ with $f\left(1\right)=0$
Bregman divergence $B_F$	$B_F(\theta :{\theta }^{\prime }):=F\left(\theta \right)-F\left({\theta }^{\prime }\right)-{(\theta -{\theta }^{\prime })}^{\top }\nabla F\left({\theta }^{\prime }\right)$
Canonical divergence $A_{F,F^{*}}$	$A_{F,F^{*}}(\theta :{\eta }^{\prime })=F\left(\theta \right)+F^{*}\left({\eta }^{\prime }\right)-{\theta }^{\top }{\eta }^{\prime }$
Bhattacharyya distance	$B_{\alpha }[p_1:p_2]=-log{\int }_{x\in \mathcal{X}}{p}_1^{\alpha }\left(x\right)p_2^{1-\alpha }\left(x\right)\mathrm{d}\mu \left(x\right)$
Jensen/Burbea-Rao divergence	$J_F^{\left(\alpha \right)}({\theta }_1:{\theta }_2)=\alpha F\left({\theta }_1\right)+(1-\alpha )F\left({\theta }_2\right)-F({\theta }_1+(1-\alpha ){\theta }_2)$
Chernoff information	$C[P_1,P_2]=-log{min}_{\alpha \in (0,1)}{\int }_{x\in \mathcal{X}}{p}_1^{\alpha }\left(x\right)p_2^{1-\alpha }\left(x\right)\mathrm{d}\mu \left(x\right)$
F, $F^{*}$	Potential functions related by Legendre–Fenchel transformation
$D_{\rho }(p,q)$	Riemannian distance $D_{\rho }(p,q):={\int }_0^1{\parallel {\gamma }^{\prime }\left(t\right)\parallel }_{\gamma \left(t\right)}\mathrm{d}t$
B, $B^{*}$	basis, reciprocal basis
$B=\{e_1={\partial }_1,\dots ,e_D={\partial }_D\}$	natural basis
${\left\{\mathrm{d}{x}_i\right\}}_i$	covector basis (one-forms)
${\left(v\right)}_B:=\left(v^i\right)$	contravariant components of vector v
${\left(v\right)}_{B^{*}}:=\left(v_i\right)$	covariant components of vector v
$u\perp v$	vector u is perpendicular to vector v ($\langle u,v\rangle =0$)
$\parallel v\parallel =\sqrt{\langle v,v\rangle }$	induced norm, length of a vector v
M, S	Manifold, submanifold
$T_p$	tangent plane at p
$TM$	Tangent bundle $TM={\cup }_p{T}_p=\{(p,v),p\in M,v\in T_p\}$
$\mathfrak{F}\left(M\right)$	space of smooth functions on M
$\mathfrak{X}\left(M\right)=\mathsf{\Gamma }\left(TM\right)$	space of smooth vector fields on M
$vf$	direction derivative of f with respect to vector v
$X,Y,Z\in \mathfrak{X}\left(M\right)$	Vector fields
$g\stackrel{\mathsf{\Sigma }}{=}{g}_{ij}\mathrm{d}{x}_i\otimes \mathrm{d}{x}_j$	metric tensor (field)
$(\mathcal{U},x)$	local coordinates x in a chat $\mathcal{U}$
${\partial }_i:=:\frac{\partial }{\partial x_i}$	natural basis vector
${\partial }^i:=:\frac{\partial }{\partial x^i}$	natural reciprocal basis vector
∇	affine connection
${\nabla }_{X}Y$	covariant derivative
${\prod }_c^{\nabla }$	parallel transport of vectors along a smooth curve c
${\prod }_c^{\nabla }v$	Parallel transport of $v\in T_{c\left(0\right)}$ along a smooth curve c
$\gamma$, ${\gamma }_{\nabla }$	geodesic, geodesic with respect to connection ∇
${\mathsf{\Gamma }}_{ij,l}$	Christoffel symbols of the first kind (functions)
${\mathsf{\Gamma }}_{ij}^k$	Christoffel symbols of the second kind (functions)
R	Riemann–Christoffel curvature tensor
$[X,Y]$	Lie bracket $[X,Y]\left(f\right)=X\left(Y\right(f\left)\right)-Y\left(X\right(f\left)\right),\forall f\in \mathfrak{F}\left(M\right)$
∇-projection	$P_S=arg{min}_{Q\in S}D(\theta \left(P\right):\theta \left(Q\right))$
${\nabla }^{*}$-projection	$P_S^{*}=arg{min}_{Q\in S}D(\theta \left(Q\right):\theta \left(P\right))$
C	Amari–Chentsov totally symmetric cubic 3-covariant tensor
$\mathcal{P}={\left\{p_{\theta }\left(x\right)\right\}}_{{\theta }_{i}n\mathsf{\Theta }}$	parametric family of probability distributions
$\mathcal{E},\mathcal{M},{\Delta }_D$	exponential family, mixture family, probability simplex
${}_{\mathcal{P}}I\left(\theta \right)$	Fisher information matrix of family $\mathcal{P}$
${}_{\mathcal{P}}I\left(\theta \right)$	Fisher Information Matrix (FIM) for a parametric family $\mathcal{P}$
${}_{\mathcal{P}}g$	Fisher information metric tensor field
exponential connection ${}_{\mathcal{P}}^e\nabla$	${}_{\mathcal{P}}^e\nabla :=E_{\theta }\left[\left({\partial }_i{\partial }_{j}l\right)\left({\partial }_{k}l\right)\right]$
mixture connection ${}_{\mathcal{P}}^m\nabla$	${}_{\mathcal{P}}^m\nabla :=E_{\theta }\left[({\partial }_i{\partial }_{j}l+{\partial }_{i}l{\partial }_{j}l)\left({\partial }_{k}l\right)\right]$
expected skewness tensor $C_{ijk}$	$C_{ijk}:=E_{\theta }\left[{\partial }_{i}l{\partial }_{j}l{\partial }_{k}l\right]$
expected $\alpha$-connections	${}_{\mathcal{P}}{{\mathsf{\Gamma }}^{\alpha }}_{ij}^k:=-\frac{1+\alpha }{2}{C}_{ijk}=E_{\theta }\left[\left({\partial }_i{\partial }_{j}l+\frac{1-\alpha }{2}{\partial }_{i}l{\partial }_{j}l\right)\left({\partial }_{k}l\right)\right]$
≡	equivalence of geometric structures