$(M,g)$	Riemannian manifold
$(\mathcal{P},{}_{\mathcal{P}}g)$	Fisher–Riemannian (expected) Riemannian manifold
$(M,g,\nabla )$	Riemannian manifold $(M,g)$ with affine connection ∇
$(\mathcal{P},{}_{\mathcal{P}}g,{}_{\mathcal{P}}{}^e{\nabla }^{\alpha })$	Chentsov’s manifold with affine exponential $\alpha$-connection
$(M,g,\nabla ,{\nabla }^{*})$	Amari’s dualistic information manifold
$(\mathcal{P},{}_{\mathcal{P}}g,{}_{\mathcal{P}}{\nabla }^{-\alpha },{}_{\mathcal{P}}{\nabla }^{\alpha })$	Amari’s (expected) information $\alpha$-manifold, $\alpha$-geometry
$(M,g,C)$	Lauritzen’s statistical manifold [29]
$(M,{}^{D}g,{}^D\nabla ,{}^{D^{*}}\nabla )$	Eguchi’s conjugate connection manifold induced by divergence D
$(M,{}^{F}g,{}^{F}C)$	Chentsov/Amari’s dually flat manifold induced by convex potential F