scholarly journals Optimizations on Statistical Hypersurfaces with Casorati Curvatures

2021 ◽  
Vol 45 (03) ◽  
pp. 449-463
Author(s):  
ALIYA NAAZ SIDDIQUI ◽  
MOHAMMAD HASAN SHAHID
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Real Hypersurface ◽  
Holomorphic Sectional Curvature ◽  
Statistical Manifold ◽  
Equality Case ◽  
Normalized Scalar Curvature

In the present paper, we study Casorati curvatures for statistical hypersurfaces. We show that the normalized scalar curvature for any real hypersurface (i.e., statistical hypersurface) of a holomorphic statistical manifold of constant holomorphic sectional curvature k is bounded above by the generalized normalized δ−Casorati curvatures and also consider the equality case of the inequality. Some immediate applications are discussed.

A lower bound of normalized scalar curvature for the submanifolds of locally conformal Kaehler space form using Casorati curvatures

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4925-4932 ◽  
Author(s):  
Mehraj Lone
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Space Form ◽  
Equality Case ◽  
Locally Conformal ◽  
Normalized Scalar Curvature

In the present paper, we prove the inequality between the normalized scalar curvature and the generalized normalized ?-Casorati curvatures for the submanifolds of locally conformal Kaehler space form and also consider the equality case of the inequality.

scholarly journals CR-submanifolds of a locally conformal Kaehler space form

1994 ◽  
Vol 17 (3) ◽  
pp. 511-514
Author(s):  
M. Hasan Shahid
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Space Form ◽  
Holomorphic Sectional Curvature ◽  
Totally Geodesic ◽  
Locally Conformal ◽  
Cr Submanifolds

(Bejancu [1,2]) The purpose of this paper is to continue the study ofCR-submanifolds, and in particular of those of a locally conformal Kaehler space form (Matsumoto [3]). Some results on the holomorphic sectional curvature,D-totally geodesic,D1-totally geodesic andD1-minimalCR-submanifolds of locally conformal Kaehler (1.c.k.)-space fromM¯(c)are obtained. We have also discussed Ricci curvature as well as scalar curvature ofCR-submanifolds ofM¯(c).

scholarly journals Bounds for generalized normalized δ-Casorati curvatures for Bi-slant submanifolds in T-space forms

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 329-340 ◽  
Author(s):  
Mohd. Aquib
Keyword(s):  
Scalar Curvature ◽  
Space Forms ◽  
Equality Case ◽  
Slant Submanifolds ◽  
Invariant Submanifolds ◽  
Normalized Scalar Curvature ◽  
Cr Submanifolds

In this paper, we prove the inequality between the generalized normalized ?-Casorati curvatures and the normalized scalar curvature for the bi-slant submanifolds in T-space forms and consider the equality case of the inequality. We also develop same results for semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds, slant submanifolds, invariant and anti-invariant submanifolds in T-space forms.

scholarly journals Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 259 ◽  
Author(s):  
Chul Lee ◽  
Jae Lee
Keyword(s):  
Scalar Curvature ◽  
Riemannian Metric ◽  
Statistical Manifold ◽  
Statistical Structure ◽  
Slant Submanifolds ◽  
Statistical Manifolds ◽  
Conjugate Connections ◽  
Levi Civita Connection ◽  
Normalized Scalar Curvature ◽  
Totally Real

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and ∇ * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant φ -sectional curvature. In addition, we give examples to show that the total space is a sphere.

scholarly journals Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 251 ◽  
Author(s):  
Simona Decu ◽  
Stefan Haesen ◽  
Leopold Verstraelen
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Holomorphic Sectional Curvature ◽  
Holomorphic Curvature ◽  
Statistical Manifolds ◽  
Casorati Curvature

In this paper, we prove some inequalities in terms of the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

scholarly journals Complete hypersurfaces in cylinders

1989 ◽  
Vol 46 (2) ◽  
pp. 313-318
Author(s):  
Thomas Hasanis
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Coordinate Functions ◽  
Bounded Below ◽  
Complete Hypersurfaces

AbstractWe consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

Discs area-minimizing in mean convex Riemannian n-manifolds

2021 ◽  
pp. 1-22
Author(s):  
Ezequiel Barbosa ◽  
Franciele Conrado
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Higher Dimensions ◽  
Dimensional Torus ◽  
Rigidity Result ◽  
Equality Case ◽  
Totally Geodesic ◽  
Zero Degree ◽  
Area Minimizing ◽  
Mean Convex

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.

Generalized -Einstein Real Hypersurfaces in and

2020 ◽  
Vol 63 (4) ◽  
pp. 909-920
Author(s):  
Yaning Wang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Coefficient ◽  
Real Hypersurface ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Real Hypersurfaces

AbstractIn this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

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