Structures on Manifolds

The condition for the curvature of a statistical manifold 
to admit a kind of standard hypersurface is given 
as a first step of the statistical submanifold theory. 
A complex version of the notion of statistical structures 
is also introduced.

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Hypersurfaces in Statistical Manifolds

We introduce lightlike hypersurfaces of a golden semi-Riemannian manifold. We investigate several properties of lightlike hypersurfaces of a golden semi-Riemannian manifold. We prove that there is no radical anti-invariant lightlike hypersurface of a golden semi-Riemannian manifold. In particular, we obtain some results for screen semi-invariant lightlike hypersurfaces of a golden semi-Riemannian manifold. Moreover, we study screen conformal screen semi-invariant lightlike hypersurfaces.

Lightlike Hypersurfaces of a Golden Semi-Riemannian Manifold

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Recent Developments in Wintgen Inequality and Wintgen Ideal Submanifolds

How do deep neural networks benefit from a very high dimensional parameter space? Their high complexity vs stunning generalization performance forms an intriguing paradox. We took an information-theoretic approach. We find that the locally varying dimensionality of the parameter space can be studied by the discipline of singular semi-Riemannian geometry. We adapt Fisher information to this singular neuromanifold. We use a new prior to interpolate between Jeffreys' prior and the Gaussian prior. We derive a minimum description length of a deep learning model, where the spectrum of the Fisher information matrix plays a key role to reduce the model complexity.

Lightlike Neuromanifolds, Occam's Razor and Deep Learning

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

https://www.mdpi.com/2227-7390/7/9/797/pdf

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

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 anti-invariant lightlike hypersurfaces. We consider lightlike hypersurfaces with respect to a quarter-symmetric nonmetric connection which is determined by the product structure. We give some equivalent conditions for integrability of distributions with respect to the Levi-Civita connection of semi-Riemannian manifolds and the quarter-symmetric nonmetric connection, and we obtain some results.

/pdf/lightlike-hypersurfaces-of-a-semi-riemannian-product-4u9etouqij.pdf

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

Lightlike hypersurfaces of a statistical manifold are studied. It is shown that a lightlike hypersurface of a statistical manifold is not a statistical manifold with respect to the induced connections, but the screen distribution has a canonical statistical structure. Some relations between induced geometric objects with respect to dual connections in a lightlike hypersurface of a statistical manifold are obtained. An example is presented. Induced Ricci tensors for lightlike hypersurface of a statistical manifold are computed.

Geometry of lightlike hypersurfaces of a statistical manifold

In this paper, we study slant submanifolds of Riemannian manifolds with Golden structure. A Riemannian manifold $(\tilde{M},\tilde{g},{\varphi})$ is called a Golden Riemannian manifold if the $(1,1)$ tensor field ${\varphi}$ on $\tilde{M}$ is a golden structure, that is ${\varphi}^{2}={\varphi}+I$ and the metric $\tilde{g}$ is ${\varphi}-$ compatible. First, we get some new results for submanifolds of a Riemannian manifold with Golden structure. Later we characterize slant submanifolds of a Riemannian manifold with Golden structure and provide some non-trivial examples of slant submanifolds of Golden Riemannian manifolds.

https://www.ijmex.com/index.php/ijmex/article/download/985/407

Slant submanifolds of golden riemannian manifolds

Kenmotsu geometry is a valuable part of contact geometry with nice applications in other fields such as theoretical physics. In this article, we study the statistical counterpart of a Kenmotsu manifold, that is, Kenmotsu statistical manifold with some related examples. We investigate some statistical curvature properties of Kenmotsu statistical manifolds. We prove that a Kenmotsu statistical manifold is not a Ricci-flat statistical manifold with an example. Finally, we prove a very well-known Chen-Ricci inequality for statistical submanifolds in Kenmotsu statistical manifolds of constant $\phi-$sectional curvature by adopting optimization techniques on submanifolds. This article ends with some concluding remarks.

Oğuzhan Bahadır

Papers

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

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

Geometry of lightlike hypersurfaces of a statistical manifold

Slant submanifolds of golden riemannian manifolds

Extremities for statistical submanifolds in Kenmotsu statistical manifolds