Related papers: K\"ahlerian information geometry for signal proces…
Information geometry provides differential geometric concepts like a Riemannian metric, connections and covariant derivatives on spaces of probability distributions. We discuss here how these concepts apply to quantum field theories in the…
We show how certain topological properties of co-K\"ahler manifolds derive from those of the K\"ahler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-K\"ahler manifold reduces the…
Two geometrical structures have been extensively studied for a manifold of probability distributions. One is based on the Fisher information metric, which is invariant under reversible transformations of random variables, while the other is…
The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K\"ahler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very…
In this paper, we introduce \emph{$\ell^p$-information geometry}, an infinite-dimensional framework that shares key features with the geometry of the space of probability densities \( \mathrm{Dens}(M) \) on a closed manifold, while also…
We associate to the resolved conifold an affine special K\"{a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\"{a}hler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show…
In this paper a class of dynamical systems describing deterministic neural network models are formulated from a viewpoint of differential geometry. This class includes the Hopfield model and gradient systems, and is such that the so-called…
We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between…
We study almost K\"ahler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ${\cal{AK}}_2$). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this…
For compact K\"ahlerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry…
Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose,…
In this paper, we consider a non-degenerate CR manifold (M,H(M),J) with a given pseudo-Hermitian 1-form {\theta}, and endow the CR distribution H(M) with any Hermitian metric h instead of the Levi form L_{{\theta}}. This induces a natural…
We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type…
We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact K\"ahler manifold. We show that if the K\"ahler potential is real analytic, then the Bergman kernel accepts a…
In recent years, there are many progress made in K\"ahler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal K\"ahler metrics, as well as obstructions to the existence of such metrics in…
We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of…
Amari's Information Geometry is a dually affine formalism for parametric probability models. The literature proposes various nonparametric functional versions. Our approach uses classical Weyl's axioms so that the affine velocity of a…
We describe the recently established minimal model program for (non-algebraic) K\"ahler threefolds as well as the abundance theorem for these spaces.
Being infinite dimensional, non-parametric information geometry has long faced an "intractability barrier" due to the fact that the Fisher-Rao metric is now a functional incurring difficulties in defining its inverse. This paper introduces…
Wasserstein gradient and Hamiltonian flows have emerged as essential tools for modeling complex dynamics in the natural sciences, with applications ranging from partial differential equations (PDEs) and optimal transport to quantum…