Manifold Connections

Standard Divergence in Manifold of Dual Affine Connections

α-Connections and a Symmetric Cubic Form on a Riemannian Manifold

In this paper, we study the construction of α-conformally equivalent statistical manifolds for a given symmetric cubic form on a Riemannian manifold. In particular, we describe a method to obtain α-conformally equivalent connections from the relation between tensors and the symmetric cubic form.

A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇(α) on a statistical manifold M equipped with a pair of conjugate connections ∇ ≡ ∇(1) and ∇∗ ≡ ∇(−1) is given as ∇(α) = 1+α 2 ∇ + 1−α 2 ∇∗. Here, we develop an expression of curvature R for ∇(α) in relation to those for ∇,∇∗. Immediately evident from it is that ∇(α) is equiaffine for any α ∈ R when ∇,∇∗ are dually flat, as previously observed in Takeuchi and Amari ...

The Volume of the Moduli Space of Flat Connections on a Nonorientable 2-manifold

We compute the Riemannian volume of the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten’s volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten’s volume formula for the moduli space of flat connections on a Riemann surface using Haar measure. ———————–

Lightlike Hypersurfaces of a Semi-Riemannian Product Manifold and Quarter-Symmetric Nonmetric Connections

We study lightlike hypersurfaces of a semi-Riemannian product manifold. We introduce a class of lightlike hypersurfaces called screen semi-invariant lightlike hypersurfaces and radical antiinvariant lightlike hypersurfaces. We consider lightlike hypersurfaces with respect to a quartersymmetric nonmetric connection which is determined by the product structure. We give some equivalent conditions ...