In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...

We study the complex hypersurfaces $f:M^{\left(n\right)}\to {ℂ}^{n+1}$ which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of ${ℂ}^{n+1}$.

We prove that every compact, normal Riemannian homogeneous manifold admits an adapted complex structure on its entire tangent bundle.

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus $T$ with a complex line $ℂ$, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to $T\times {\mathbb{P}}^1$. The converse statement is also true, namely a holomorphic bundle on $T\times {\mathbb{P}}^1$ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....