Statistical manifolds with almost contact structures and its statistical submersions

In this paper, we discuss statistical manifolds with almost contact sturctures. We define a Sasaki-like statistical manifold. Moreover, we consider Sasaki-like statistical submersions, and we study Sasaki-like statistical submersion with the property that the curvature tensor with respect to the affine connection of the total space satisfies the condition (2.12).     

Statistical manifolds are statistical models

In this note we prove that any smooth (C¹ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers to Lauritzen’s question and Amari’s question on a realization of smooth (C¹ resp.) statistical manifolds as finite dimensional statistical models.     

Equiaffine structures on statistical manifolds and Bayesian statistics

Relations between equiaffine geometry and Bayesian statistics are studied. A prior distribution in Bayesian statistics is regarded as a volume form on a statistical manifold. Applying equiaffine geometry to Bayesian statistics, the relation between alpha-parallel priors and the Jeffreys prior is given. As geometric results, conditions for a statistical submanifold to have an equiaffine structure are also given.     

Finite-dimensional algebraic representations of the infinite phylon group

The concept of phylon is introduced as a generalisation of derivative strings, differential strings and new tensors. The behaviour of phyla under change of coordinates is given by finite-dimensional algebraic representations of a very large group, the infinite phylon group. These representations are studied from both the general and the matrix points of view. Various examples of phyla are given, mainly from a statistical context. The basic structure of these representations is given.     

The geometric structures and instability of entropic dynamical models

In this paper, we characterize two entropic dynamical (ED) models from the viewpoint of information geometry and give the geometric structures of the associated statistical manifolds of the models. The scalar curvatures and the geodesics are obtained. Also the instability of entropic dynamical models is studied from the behavior of the geodesics lengths, statistical volume elements and Jacobi vector fields.     

On \alpha-conformal equivalence of statistical submanifolds

In this paper, we show a procedure to realize a statistical manifold, which is -conformally equivalent to a manifold with an -transitively flat connection, as a statistical submanifold. Received 29 December 1999.     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds

In this paper, we study pseudo-slant submanifolds and their warped products in Kenmotsu manifolds. We obtain the necessary conditions that a pseudoslant submanifold is locally a warped product and establish an inequality for the squared norm of the second fundamental form in terms of the warping function. The equality case is also considered.     

Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-K?hler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-K?hler is given.      

Globally minimal homogeneous subspaces in compact homogeneous sympletic spaces

It is a general problem to describe and classify the globally minimal surfaces in homogeneous spaces. The present paper studies and answers the following problem: When is a homogeneous subspace whose isometry group is one of the classical groups, a globally minimal submanifold in a regular orbit of the adjoint representation of a classical group?     

Ricci flat metrics with bidimensional null orbits and non-integrable orthogonal distribution

We study Ricci flat 4-metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature (++−−) and, moreover, homogeneous possessing a 6-dimensional symmetry algebra.     

Isotropic ppmc immersions

Isometric immersions with parallel pluri-mean curvature (“ppmc”) in euclidean n-space generalize constant mean curvature (“cmc”) surfaces to higher dimensional Kähler submanifolds. Like cmc surfaces they allow a one-parameter family of isometric deformations rotating the second fundamental form at each point. If these deformations are trivial the ppmc immersions are called isotropic. Our main result drastically restricts the intrinsic geometry of such a submanifold: Locally, it must be a symmetric space or a Riemannian product unless the immersion is holomorphic or a superminimal surface in a sphere. We can give a precise classification if the codimension is less than 7. The main idea of the proof is to show that the tangent holonomy is restricted and to apply the Berger-Simons holonomy theorem.     

Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

Compact Hermitian surfaces of Einstein type with respect to the Hermitian connection

Two Einstein-type conditions for the Hermitian curvature tensor are considered on a compact Hermitian surface: It is proved that if the symmetric part of the Ricci tensors is a scalar multiple of the metric with a negative constant, then the metric is Kaehler. If the Hermitian surface satisfies the Hermite-Einstein condition with a non positive constant, then the metric is Kaehler.Supported by Contract MM 413/1994 with the Ministry of Science and Education of Bulgaria.Supported by Contract MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.     

Extrinsic homogeneity of parallel submanifolds II

We discuss the question whether a (complete) parallel submanifold M of a Riemannian symmetric space N is an (extrinsically) homogeneous submanifold, i.e. whether there exists a subgroup of the isometries of N which acts transitively on M. In a previous paper, we have discussed this question in case the universal covering space of M is irreducible. It is the subject of this paper to generalize this result to the case when the universal covering space of M has no Euclidian factor.     

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.     

Pappus type theorems for motions along a submanifold

We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form Mⁿ_λ. We show: (a) volume(D) depends only on the second fundamental form of P, whereas volume(C) depends on all the ith fundamental forms of P, (b) when the domain that we move D₀ has its q-centre of mass on P, volume(D) does not depend on the mean curvature of P, (c) when D₀ is q-symmetric, volume(D) depends only on the intrinsic curvature tensor of P; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(n−q−d), and C is closed, then volume(C) does not depend on the ith fundamental forms of P for i>2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (n−q)-plane orthogonal to P) around a d-dimensional geodesic plane.     

Conformal vector fields with respect to the Sasaki metric tensor

The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact, it is found that the only conformal vector fields are Killing vector fields.