Abstract
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. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces.
Highlights
The name “statistical geometry” is relatively new, this geometry has existed for long and in various editions
Starting from locally strongly convex hypersurfaces in the Euclidean space, through locally strongly convex equiaffine hypersurfaces in the affine space Rn+1 and Lagrangian submanifolds in complex space forms to Hessian manifolds—all these examples are statistical manifolds
If a statistical structure can be realized on a Lagrangian submanifold and on an affine hypersurface, its Riemannian sectional curvature, its ∇-sectional curvature, and its K -sectional curvature are all constant
Summary
The name “statistical geometry” is relatively new, this geometry has existed for long and in various editions. A statistical structure is nothing but a Codazzi pair. Starting from locally strongly convex hypersurfaces in the Euclidean space, through locally strongly convex equiaffine hypersurfaces in the affine space Rn+1 and Lagrangian submanifolds in complex space forms to Hessian manifolds—all these examples are statistical manifolds. Note that the structures of the subclasses have very different properties and, the intersections of the subclasses are small. If a statistical structure can be realized on a Lagrangian submanifold and on an affine hypersurface, its Riemannian sectional curvature, its ∇-sectional curvature, and its K -sectional curvature are all constant. Note that the category of statistical structures is much larger than the union of all the specific subclasses mentioned above. Results proved for affine hypersurfaces or for Lagrangian submanifolds usually are rarely generalizable to the general case of statistical structures
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