We review the information-geometric framework for statistical pattern recognition: First, we explain the role of statistical similarity measures and distances in fundamental statistical pattern recognition problems. We then concisely review the main statistical distances and report a novel versatile family of divergences. Depending on their intrinsic complexity, the statistical patterns are lea...

Information geometry is an important tool to study statistical models. There are some examples in models which regarded as warped products. In this paper, we information of We consider the case where product and its fiber space equipped with dually flat connections and, particular a cone, characterize on base $$\mathbb {R}_{>0}$$ . The resulting turn out be $$\alpha $$ -connections = \pm {1}$$

Let K denote a closed odd-dimensional smooth manifold and let E be a flat vector bundle over K. In this situation the construction of Ray and Singer [RS] gives a metric on the determinant line of the cohomology detH(M ;E) which is a smooth invariant of the manifold M and the flat bundle E. (Note that if the dimension of K is even then the Ray-Singer metric depends on the choice of a Riemannian ...

We show that the non flat 2+1 dimensional factor of Gödel’s metric can be seen as belonging to a one parameter family of squashed anti–de Sitter geometries, with or without causal pathologies. We give a global algebraic isometric embedding of these metrics in 4+3 or 3+4 dimensional flat spaces, thereby illustrating the occurrence of the closed timelike curves. Mâıtre de Recherches F.N.R.S. E-ma...

In this paper, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. 1. The result A complete Riemannian metric gij on a smooth manifold M n is called a gradient steady Ricci soliton if there exists a smooth function F on M such that the Ricci tensor Rij of the metric gij is given by the Hessian of F : Rij = ∇i∇jF. (1....

We introduce a very natural class of potential submanifolds in pseudo-Euclidean spaces (each Ndimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudoEuclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanifold. We show that all potential submanifolds bear natural special ...

in the first part of this paper, some theorems are given for a riemannian manifold with semi-symmetric metric connection. in the second part of it, some special vector fields, for example, torse-forming vector fields, recurrent vector fields and concurrent vector fields are examined in this manifold. we obtain some properties of this manifold having the vectors mentioned above.

Locations and patterns of functional brain activity in humans are difficult to compare across subjects because of differences in cortical folding and functional foci are often buried within cortical sulci. Unfolding a cortical surface via flat mapping has become a key method for facilitating the recognition of new structural and functional relationships. Mathematical and other issues involved i...

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4, 5, 6, 7 and 8. It is shown that any n-dimensional (4 ≤ n ≤ 8)...