An affine submersion with horizontal distribution and its applications

Vertical martingales, stochastic calculus and harmonic sections

“…Let H p = (π p | HpE ) −1 : T x M → H p E be the horizontal lift, where π(p) = x. We observe that H p is an isomorphism for each p ∈ E. The submersion π : E → M is called affine submersion with horizontal distribution if h∇ E H(X) H(Y ) = H(∇ M X Y ) (see [1] for more details). A Riemmanian submersion is a classical example of affine submersion with horizontal distribution.…”

Statistical manifolds with almost contact structures and its statistical submersions

“…An almost complex structure on a manifold M is a tensor field ϕ of type (1,1) such that ϕ 2 = −I, where I stands for the identity transformation. An almost complex manifold is such a manifold with a fixed almost complex structure.…”

“…A semi-Riemannian submersion π : M → B is a submersion such that all fibers are semi-Riemannian submanifolds of M , and π * preserves lengths of horizontal vectors ( [12]). Recently, N. Abe and K. Hasegawa ( [1]) studied an affine submersion with horizontal distribution. They investigated when the total space is the statistical manifold.…”

Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

“…(see, e.g., [27][28][29][30][31][32]). In particular, the differential geometry field is focused on topics such as submanifold theory of statistical manifolds [33], Hessian geometry [34], statistical submersions [35], complex manifold theory of statistical manifolds ( [29,36,37]), contact theory on statistical manifolds [38], and quaternionic theory on statistical manifolds [39]. For the above problems, Aydin et al obtained Chen-Ricci inequalities [40] and a generalized Wintgen inequality [41] for submanifolds in statistical manifolds of constant curvature.…”