Curvature bounded conjugate symmetric statistical structures with complete metric

Abstract

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 sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.1991 Mathematics Subject Classification. Primary: 53C05, 53C20 53A15, 53C21.

Some Inequalities and Applications of Simons’ Type Formulas in Riemannian, Affine, and Statistical Geometry

“…More generally, if for a statistical structure Ric = Ric , then dτ = 0. Indeed, by writing (15) for ∇ and comparing with (15), we see that Ric = Ric if and only if div K = ∇τ . Therefore, if Ric = Ric , then dτ = 0 (because div K is symmetric).…”

“…Therefore, if Ric = Ric , then dτ = 0 (because div K is symmetric). Using (15) and the analogous formula for the connection ∇, one also gets…”

Some inequalities and applications of Simons' type formulas in Riemannian, affine and statistical geometry

“…By Theorem 2.1, we now have that statistical structures of constant curvature are (from a local viewpoint) exactly the induced structures on equiaffine spheres. Let us rewrite (15) in the case where R = HR 0 :…”

“…The results contained in the following theorem and their generalizations can be found in [1], [10], [15]. Theorem 5.4.…”

Completness of Statistical Structures

“…Let t = ϕ(s) be an affine parameter of our geodesic, say γ(t). We can assume that ϕ ′ > 0 on R. Then γ(ϕ(s)) = r(s), ϕ ′ ( γ • ϕ) = ṙ and consequently (10) ∇…”