Abstract
We begin the study of completeness of affine connections, especially those on statistical manifolds or on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
Highlights
Affine hypersurfaces have been studied for more than 100 years, the completeness of affine connections naturally appearing on such hypersurfaces was considered, to the knowledge of the author, only in [7] and [8]
In the literature of affine differential geometry, the affine completeness is always meant as the completeness of the affine metric
It is worth to note that the most beautiful results dealing with completeness of the Blaschke metric were proved many years after inventing affine differential geometry
Summary
Affine hypersurfaces have been studied for more than 100 years, the completeness of affine connections naturally appearing on such hypersurfaces was considered, to the knowledge of the author, only in [7] and [8]. The aim of this paper is to initiate the study of completeness of affine connections appearing in the theory of hypersurfaces as well as in more general situations. We observe that on an affine hypersurface with parallel cubic form the induced connection is not complete unless the induced structure is trivial. We prove this result in a more general setting, namely for statistical structures on abstract manifolds and with the assumption weaker than that about parallel cubic form. The class of hypersurfaces with parallel cubic form is rich of examples and important in affine differential geometry. The geometry of affine hypersurfaces is much more developed than the geometry of statistical manifolds It provides a lot of examples and allows to use the reach technique of the induced objects which makes the considerations more imaginable.
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