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14 pages, 268 KiB

Open AccessArticle

Ricci–Yamabe Solitons on Sasakian Manifolds with the Generalized Tanaka–Webster Connection

by Abdul Haseeb

AppliedMath 2025, 5(1), 22; https://doi.org/10.3390/appliedmath5010022 - 3 Mar 2025

Viewed by 556

In this article, we analyze some curvature restrictions satisfying by the concircular curvature tensor in

(2 n + 1)

-dimensional Sasakian manifolds with the generalized Tanaka–Webster connection

\bar{\nabla}

admitting Ricci–Yamabe solitons. Finally, we give an example of three-dimensional Sasakian manifolds [...] Read more.

In this article, we analyze some curvature restrictions satisfying by the concircular curvature tensor in

(2 n + 1)

-dimensional Sasakian manifolds with the generalized Tanaka–Webster connection

\bar{\nabla}

admitting Ricci–Yamabe solitons. Finally, we give an example of three-dimensional Sasakian manifolds which verifies some of our findings. Full article

(This article belongs to the Special Issue Advances in Iterative Methods and Stability Analysis for Solving Nonlinear Problems)

82 pages, 748 KiB

Open AccessArticle

C-R Immersions and Sub-Riemannian Geometry

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Axioms 2023, 12(4), 329; https://doi.org/10.3390/axioms12040329 - 28 Mar 2023

Cited by 1 | Viewed by 2207

Abstract

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form

θ

, we consider natural

ϵ

-contractions, i.e., contractions

g_{ϵ}^{M}

of the Levi form

G_{θ}

, such that the norm [...] Read more.

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form

θ

, we consider natural

ϵ

-contractions, i.e., contractions

g_{ϵ}^{M}

of the Levi form

G_{θ}

, such that the norm of the Reeb vector field T of

(M, θ)

is of order

O (ϵ^{- 1})

. We study isopseudohermitian (i.e.,

f^{*} Θ = θ

) Cauchy–Riemann immersions

f : M \to (A, Θ)

between strictly pseudoconvex CR manifolds M and A, where

Θ

is a contact form on A. For every contraction

g_{ϵ}^{A}

of the Levi form

G_{Θ}

, we write the embedding equations for the immersion

f : M \to (A, g_{ϵ}^{A})

. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as

ϵ \to 0^{+}

. For every isopseudohermitian immersion

f : M \to S^{2 N + 1}

into a sphere

S^{2 N + 1} \subset C^{N + 1}

, we show that Webster’s pseudohermitian scalar curvature R of

(M, θ)

satisfies the inequality

R \leq 2 n [(f^{*} g_{Θ}) (T, T) + n + 1] + \frac{1}{2} {∥ H (f) ∥_{g_{Θ}^{f}}^{2} + ∥ {trace}_{G_{θ}} Π_{H (M)} (\nabla^{⊤} - \nabla) ∥_{f^{*} g_{Θ}}^{2}}

with equality if and only if

B (f) = 0

and

\nabla^{⊤} = \nabla

H (M) \otimes H (M)

. This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

40 pages, 463 KiB

Open AccessArticle

Beltrami Equations on Rossi Spheres

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Mathematics 2022, 10(3), 371; https://doi.org/10.3390/math10030371 - 25 Jan 2022

Viewed by 2259

Abstract

Beltrami equations

{\bar{L}}_{t} (g) = μ (\cdot, t) L_{t} (g)

S^{3}

(where

L_{t}

| t | < 1

, are the Rossi operators i.e.,

L_{t}

spans the globally [...] Read more.

Beltrami equations

{\bar{L}}_{t} (g) = μ (\cdot, t) L_{t} (g)

S^{3}

(where

L_{t}

| t | < 1

, are the Rossi operators i.e.,

L_{t}

spans the globally nonembeddable CR structure

H (t)

S^{3}

discovered by H. Rossi) are derived such that to describe quasiconformal mappings

f : S^{3} \to N \subset C^{2}

from the Rossi sphere

(S^{3}, H (t))

. Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions

g_{t}

such that

g_{t} - v \in W_{F}^{1, 2} (S^{3}, θ)

with

v \in {CR}^{\infty} (S^{3}, H (0))

. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications Part II)

13 pages, 256 KiB

Open AccessArticle

Contact Metric Spaces and pseudo-Hermitian Symmetry

by Jong Taek Cho

Mathematics 2020, 8(9), 1583; https://doi.org/10.3390/math8091583 - 14 Sep 2020

Viewed by 3136

Abstract

We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold

M^{2 n + 1}

n \geq 2

, is locally pseudo-Hermitian symmetric and satisfies

\nabla_{ξ} h = μ h ϕ

μ \in R

, if and only if M [...] Read more.

We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold

M^{2 n + 1}

n \geq 2

, is locally pseudo-Hermitian symmetric and satisfies

\nabla_{ξ} h = μ h ϕ

μ \in R

, if and only if M is either a Sasakian locally

ϕ

-symmetric space or a non-Sasakian

(k, μ)

-space. When

n = 1

, we prove a classification theorem of contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry. Full article

(This article belongs to the Special Issue Complex and Contact Manifolds)

12 pages, 285 KiB

Open AccessArticle

A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

by George Kaimakamis, Konstantina Panagiotidou and Juan de Dios Pérez

Mathematics 2020, 8(4), 642; https://doi.org/10.3390/math8040642 - 21 Apr 2020

Cited by 1 | Viewed by 2300

Abstract

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator [...] Read more.

F_{X}^{(k)}

is defined and is related to both connections. If X belongs to the maximal holomorphic distribution

D

on M, the corresponding operator does not depend on k and is denoted by

F_{X}

and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that

F_{X} S = S F_{X}

, where S denotes the Ricci tensor of M and a further condition is satisfied, are classified. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

11 pages, 775 KiB

Open AccessFeature PaperArticle

Slant Curves in Contact Lorentzian Manifolds with CR Structures

by Ji-Eun Lee

Mathematics 2020, 8(1), 46; https://doi.org/10.3390/math8010046 - 1 Jan 2020

Cited by 7 | Viewed by 2527

Abstract

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the

\hat{\nabla}

-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we [...] Read more.

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the

\hat{\nabla}

-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when

c \leq 0

, there does not exist a non-geodesic slant Frenet curve satisfying the

\hat{\nabla}

-Jacobi equations for the

\hat{\nabla}

-geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms

M_{1}^{3} (\hat{H})

for

\hat{H} = 2 c > 0

. Full article

(This article belongs to the Special Issue Sasakian Space)

12 pages, 295 KiB

Open AccessArticle

Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms

by George Kaimakamis, Konstantina Panagiotidou and Juan De Dios Pérez

Mathematics 2018, 6(5), 84; https://doi.org/10.3390/math6050084 - 20 May 2018

Cited by 1 | Viewed by 3503

Abstract

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field

P = ϕ A - A ϕ

is given and three-dimensional real hypersurfaces in non-flat complex space forms whose [...] Read more.

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field

P = ϕ A - A ϕ

is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field

P

satisfies geometric conditions are classified. Full article

(This article belongs to the Special Issue Differential Geometry)

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