Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature

M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

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Curvature of K-contact Semi-Riemannian Manifolds

Canadian Mathematical Bulletin ◽

10.4153/cmb-2013-016-7 ◽

2014 ◽

Vol 57 (2) ◽

pp. 401-412 ◽

Cited By ~ 7

Author(s):

Domenico Perrone

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Sectional Curvature ◽

Riemannian Manifolds ◽

Lorentzian Manifold ◽

Constant Sectional Curvature ◽

Conformally Flat ◽

Reeb Vector ◽

Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

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Biharmonic hypersurfaces in pseudo-Riemannian space forms

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816500948 ◽

2016 ◽

Vol 13 (07) ◽

pp. 1650094 ◽

Cited By ~ 1

Author(s):

Dan Yang ◽

Yu Fu

Keyword(s):

Sectional Curvature ◽

Mean Curvature ◽

Riemannian Space ◽

Constant Mean Curvature ◽

Space Form ◽

Constant Sectional Curvature ◽

Principal Curvatures ◽

Space Forms ◽

Riemannian Space Forms

Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

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Explicit construction of Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c)

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005783 ◽

2002 ◽

Vol 132 (3) ◽

pp. 481-508 ◽

Cited By ~ 3

Author(s):

YUN MYUNG OH

Keyword(s):

Sectional Curvature ◽

Isometric Immersion ◽

Complex Space ◽

Space Form ◽

Real Space ◽

Constant Sectional Curvature ◽

Complex Space Form ◽

Product Decomposition ◽

Twisted Product ◽

Space Forms

In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0 and c < 0.

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REILLY INEQUALITIES OF ELLIPTIC OPERATORS ON CLOSED SUBMANIFOLDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000355 ◽

2009 ◽

Vol 80 (2) ◽

pp. 335-346

Author(s):

RUSHAN WANG

Keyword(s):

Elliptic Operator ◽

Sectional Curvature ◽

Riemannian Manifolds ◽

Upper Bound ◽

Vector Fields ◽

Elliptic Operators ◽

Position Vector ◽

Space Forms ◽

Nonzero Eigenvalue ◽

Closed Hypersurfaces

AbstractUsing generalized position vector fields we obtain new upper bound estimates of the first nonzero eigenvalue of a kind of elliptic operator on closed submanifolds isometrically immersed in Riemannian manifolds of bounded sectional curvature. Applying these Reilly inequalities we improve a series of recent upper bound estimates of the first nonzero eigenvalue of the Lr operator on closed hypersurfaces in space forms.

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Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms

Journal of Function Spaces ◽

10.1155/2021/6195939 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Yanlin Li ◽

Akram Ali ◽

Fatemah Mofarreh ◽

Abimbola Abolarinwa ◽

Rifaqat Ali

Keyword(s):

Riemannian Manifolds ◽

Space Form ◽

Upper Bounds ◽

Laplacian Operator ◽

First Eigenvalue ◽

Sasakian Space Form ◽

Slant Submanifold ◽

Space Forms ◽

Slant Submanifolds ◽

Sasakian Space Forms

This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the ϕ -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the ϕ -Laplacian operator on closed oriented m -dimensional slant submanifolds in a Sasakian space form M ~ 2 k + 1 ε is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the ϕ -Laplacian on slant submanifold in a sphere S 2 n + 1 with ε = 1 and ϕ = 2 .

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An integral formula for hypersurfaces in space forms

Glasgow Mathematical Journal ◽

10.1017/s001708950003161x ◽

1995 ◽

Vol 37 (3) ◽

pp. 337-341 ◽

Cited By ~ 1

Author(s):

Theodoros Vlachos

Keyword(s):

Riemannian Manifold ◽

Sectional Curvature ◽

Distance Function ◽

Integral Formula ◽

Constant Sectional Curvature ◽

Position Vector ◽

Space Forms ◽

Image Position ◽

Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

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A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

Mathematics ◽

10.3390/math8030444 ◽

2020 ◽

Vol 8 (3) ◽

pp. 444

Author(s):

Muhittin Evren Aydin ◽

Adela Mihai

Keyword(s):

Sectional Curvature ◽

Present Note ◽

Dimensional Space ◽

Golden Ratio ◽

Gauss Curvature ◽

Constant Sectional Curvature ◽

Round Sphere ◽

Space Forms ◽

Totally Umbilical ◽

3 Dimensional

In the present note we introduce a Pythagorean-like formula for surfaces immersed into 3-dimensional space forms M 3 ( c ) of constant sectional curvature c = − 1 , 0 , 1 . More precisely, we consider a surface immersed into M 3 c satisfying I 2 + II 2 = III 2 , where I , II and III are the matrices corresponding to the first, second and third fundamental forms of the surface, respectively. We prove that such a surface is a totally umbilical round sphere with Gauss curvature φ + c , where φ is the Golden ratio.

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