Clairaut Submersion

In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.

On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n>2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.

The Rigging Technique for Null Hypersurfaces

Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null hypersurfaces, with emphasis in null cones, totally geodesic, totally umbilic, and compact null hypersurfaces. On the other hand, we show the interplay with the ambient space, including its influence in causality theory.

Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.

Discs area-minimizing in mean convex Riemannian n-manifolds

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to $\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where $\mathbb {D}^{2}$ is a disc and $T^{n}$ is an $n$ -dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.

Certain results on invariant submanifolds of an almost Kenmotsu $$(\kappa ,\mu ,\nu )$$-space

In the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Riemannian curvature tensor has $$(\kappa ,\mu ,\nu )$$ ( κ , μ , ν ) -nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions $$\kappa ,\mu $$ κ , μ and $$\nu $$ ν behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu $$(\kappa ,\mu ,\nu )$$ ( κ , μ , ν ) -space to be totally geodesic under some conditions.

Lightlike hypersurfaces in spaces with concircular fields

Lightlike hypersurfaces in semi-Riemannian manifolds admitting concircular vector fields are investigated. We prove that such hypersurfaces are generally products of lightlike curves and warped product manifolds. In special cases, we show that these hypersurfaces are totally geodesic or totally screen geodesic provided such concircular fields belong to their normal or transversal bundles. A number of examples are furnished, where possible, to illustrate the main concepts.

A β-tensor on Kaehler manifolds and its geometric characterizations

In this paper, we first introduce a new notion [Formula: see text]-tensor on Hermitian manifold and particularly, we present some geometric characterizations of such a tensor on the Kaehler manifold. Here, we investigate the Kaehler submersion whose total space is equipped with the [Formula: see text]-tensor and obtain some results. Also, we deal with a Kaehler submersion with totally geodesic fibers such that the total space admits [Formula: see text]-Ricci soliton and [Formula: see text]-tensor. Finally, we give necessary conditions for which any fiber and base manifold of Kaehler submersion is [Formula: see text]-Ricci soliton or [Formula: see text]-Kaehler-Einstein.

Coordinate-wise transformation of probability distributions to achieve a Stein-type identity

It is shown that for any given multi-dimensional probability distribution with regularity conditions, there exists a unique coordinate-wise transformation such that the transformed distribution satisfies a Stein-type identity. A sufficient condition for the existence is referred to as copositivity of distributions. The proof is based on an energy minimization problem over a totally geodesic subset of the Wasserstein space. The result is considered as an alternative to Sklar’s theorem regarding copulas, and is also interpreted as a generalization of a diagonal scaling theorem. The Stein-type identity is applied to a rating problem of multivariate data. A numerical procedure for piece-wise uniform densities is provided. Some open problems are also discussed.