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Some inequalities and applications of Simons' type formulas in Riemannian, affine and statistical geometry
Barbara Opozda
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative…
Completeness in affine and statistical geometry
Barbara Opozda
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems a…
Curvature bounded conjugate symmetric statistical structures with complete metric
Barbara Opozda
In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant section…
Affine spheres with prescribed Blaschke metric
Barbara Opozda
It is proved that the equality , where is the Gaussian curvature of a metric tensor g on a 2-dimensional manifold is a sufficient and necessary condition for loca…
Cauchy-Kowalevski's theorem applied for counting geometric structures
Barbara Opozda, Włodzimierz Mikulski
How many are linear connections with prescribed Ricci tensor? How many are statistical structures? The questions are answered in the analytic case by using the Cauchy-Kowalewski th…
Bochner's technique for statistical structures
Barbara Opozda
The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical struc…