Traceless cubic forms on statistical manifolds and Tchebychev geometry

Abstract

Abstract. Geometry of traceless cubic forms is studied. It is shown that the traceless part of the cubic form on a statistical manifold determines a conformal-projective equivalence class of statistical manifolds. This conformal-projective equivalence on statistical manifolds is a natural generalization of conformal equivalence on Riemannian manifolds. As an application, Tchebychev type immersions in centroaffine immersions of codimension two are studied.1. Introduction. The purpose of this paper is to study geometry of traceless cubic forms on statistical manifolds. Cubic forms and their traceless parts have been studied in affine differential geometry. In particular, it is known that an affine hypersurface lies on a quadratic if its cubic form vanishes everywhere on the hypersurface (e.g. [1]).In affine hypersurface theory, the traceless cubic form is invariant under choice of equiaffine transversal vector fields. (In other words, we often say that a traceless cubic form is invariant under relative normalizations.) We generalize this property, namely, we show that the traceless cubic form on a statistical manifold is invariant under conformalprojective changes of statistical manifolds. Moreover, we show that the traceless cubic form determines a conformal-projective class of statistical manifolds. The notion of conformal-projective class of statistical manifolds was introduced by the present author in [6]. This class induces conformal-projective geometry of statistical manifolds, which is a natural generalization of conformal geometry of Riemannian manifolds.Traceless cubic forms are useful objects in Tchebychev hypersurface theory, and conformal-projective relations are useful objects in geometry of centroaffine immersions

Conformal Control Tools for Statistical Manifolds and for γ-Manifolds

“…Let (M, e u g, ∇) be a (sgn( ) γ)-manifold and ∇0 the Levi-Civita connection of e u g. Denote à := ∇ − ∇0 . Suppose, moreover, the relation (18) holds. Then the manifolds (M, g, ∇) and (M, e u g, ∇) are f -conformal equivalent if, and only if,…”

“…Theorem 5 generalizes the main result from [9], which was proven in the particular case of statistical Riemannian manifolds (i.e., for = −1, γ = γ = 0 and g Riemannian metric). It provides a framework for the construction of pairs of f -conformal equivalent γ-manifolds, starting from the Levi-Civita connections ∇ 0 and ∇0 , the functions u and f , the tensor fields A and à and the cubic forms γ and γ, subject to the compatibility constraints (18) and (22).…”

“…Several generalizations of the conformal geometry of statistical manifolds were defined and studied, such as the conformal-projective geometry [14][15][16][17][18][19] and the geometry of semi-Weyl manifolds in [20].…”

“…instead of (1). The γ-manifolds simultaneously provide a generalization, an extension and an analogue of statistical manifolds, and, at the same time, a generalization of the semi-Weyl manifolds from [18] and of the statistical manifolds "with torsion" from [21,22].…”