Three-dimensional Riemannian Manifold Research Articles

Investigations on a Riemannian Manifold with a Semi- Symmetric Non-Metric Connection and Gradient Solitons

This article carries out the investigation of a three-dimensional Riemannian manifold N3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N3. It is established that a N3 with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and m-quasi Einstein solitons of gradient type, respectively.

A note on area estimates for stable H-hypersurfaces and applications

We show some area estimates for stable CMC hypersurfaces immersed in Riemannian manifolds with scalar and sectional curvature bounded from below. In particular, we focus on immersions in three-dimensional Riemannian manifolds. As an application, we derive some upper estimates for the bottom spectrum of these hypersurfaces. This paper generalizes and builds on the techniques developed by Munteanu, Sung and Wang [19] for stable minimal surfaces.

On foliations of bounded mean curvature on closed three-dimensional Riemannian manifolds

The notion of systole of a foliation sys(ℱ) on an arbitrary foliated closed Riemannian manifold (M,ℱ) is introduced. A lower bound on sys(ℱ) of a bounded mean curvature foliation is given. As a corollary we prove that the number of Reeb components of a bounded mean curvature foliation on a closed oriented Riemannian 3-manifold M is bounded above by a constant depending on the volume, the radius of injectivity, and the maximum value of the sectional curvature of the manifold M.

A split special Lagrangian calibration associated with frame vorticity

Abstract Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO ⁢ ( M ) → M {\mathrm{SO}(M)\rightarrow M} of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso o ⁢ ( M ) ≅ SO ⁢ ( M ) {\mathrm{Iso}_{o}(M)\cong\mathrm{SO}(M)} : A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

Equivariant index bound for min–max free boundary minimal surfaces

Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min–max procedure á la Simon–Smith with n-parameters is bounded above by n.

Gaussian curvature of minimal graphs in [formula omitted

In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold M×R. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via ℘-harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.

Total Mean Curvature and First Dirac Eigenvalue

In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in the Euclidean space by combining a positive mass theorem and the construction of quasi-spherical metrics. As a direct consequence of this estimate, we obtain an asymptotic expansion for the first eigenvalue of the Dirac operator on large spheres in three-dimensional asymptotically flat manifolds. We also study this expansion for small geodesic spheres in a three-dimensional Riemannian manifold. We finally discuss how this method can be adapted to yield similar results in the hyperbolic space.

Sharp Weyl laws with singular potentials

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumovi\'c to the case of Schr\"odinger operators with singular potentials.

Riemannian 3-manifolds and Ricci–Yamabe solitons

In this paper, we characterize three-dimensional Riemannian manifolds [Formula: see text] admitting Ricci–Yamabe solitons (RYSs). It is proved that if an [Formula: see text] endowed with a semi-symmetric metric [Formula: see text]-connection admits an RYS, then the scalar curvature of [Formula: see text] satisfies the Poisson equation [Formula: see text] where [Formula: see text] and [Formula: see text] We also discuss the existence of gradient RYS in Riemannian setting. Finally, we construct a nontrivial example of three-dimensional Riemannian 3-manifolds admitting RYS to prove some of our results.

Geometry of a surface in Riemannian 3-manifold corresponding to a smooth autonomous dynamical system

In this work, we identify a smooth autonomous dynamical on a two-dimensional manifold with an exterior differential system [Formula: see text], where [Formula: see text] is a three-dimensional Riemannian manifold and [Formula: see text] is the differential ideal generated by the contact forms on [Formula: see text]. We investigate the intrinsic and the extrinsic geometry of a surface in [Formula: see text] and show that for a particular dynamical system [Formula: see text] admits a totally geodesic surface determined by a constant value of a coordinate function. We also exhibit that such a surface may define intrinsically nonflat minimal surface which is not necessarily totally geodesic.

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a constant mean curvature surface with free boundary in C. We provide a pinching condition on the length of the traceless second fundamental form on $$\Sigma $$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if C is a geodesic ball of three-dimensional space forms, then $$\Sigma $$ is either a spherical cap or a Delaunay surface.

Foliations on closed three-dimensional Riemannian manifolds with small modulus of mean curvature of the leaves

We prove that the modulus of mean curvature of the leaves of a transversely oriented foliation of codimension one with a generalized Reeb component on an oriented smooth closed three-dimensional Riemannian manifold cannot be everywhere smaller than a certain positive constant depending on the volume, the maximum value of the sectional curvature, and the injectivity radius of the manifold. This means that foliations with small modulus of mean curvature of the leaves are taut.

The Curvature Property of a Linear Dynamical System

In this work a two-dimensional smooth autonomous dynamical system is regarded as a three-dimensional Riemannian manifold and it is shown that the scalar curvature of a linear dynamical system $\text{d}x/\text{d}t=ax+by$, $\text{d}y/\text{d}t=cx+dy$ is non-positive. The manifold is scalar-flat iff $b=-c$ and $a=d=0$.

Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold \((M^3,g)\) that admits a smooth nontrivial solution f to the equation $$\begin{aligned} \nabla df=\psi Rc+\phi g, \end{aligned}$$ (1) where \(\psi ,\phi \) are given smooth functions of f, Rc is the Ricci tensor of g. Spaces of this type include various interesting classes, namely gradient Ricci solitons, m-quasi Einstein metrics, (vacuum) static spaces, V-static spaces, and critical point metrics. The m-quasi Einstein metrics and vacuum static spaces were previously studied in Jordan (Gen Relativ Gravit 41(9):2191–2280, 2009) and Kim and Shin (Math Nachr 292(8): 1727–1750, 2019), respectively. In this paper, we refine them and develop a general approach for the solutions of (1). We specify the shape of the metric g satisfying (1) when \(\nabla f\) is not a Ricci-eigen vector. Then we focus on the remaining three classes, namely gradient Ricci solitons, V-static spaces, and critical point metrics. Furthermore, we present classifications of local three-dimensional Ricci-degenerate spaces of these three classes by explicitly describing the metric g and the potential function f.

Generic uniqueness and stability for the mixed ray transform

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 plus 1"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 plus 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q upper S"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">qS</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polarized elastic waves provide partial information about the mixed ray transform of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 plus 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the \text{Dirichlet-to-Neumann} map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

A Note on Gradient Solitons in Three-Dimensional Riemannian Manifolds

We charecterize three-dimensional Riemannian manifolds endowed with a special type of vector field if the Riemannian metrices are gradient Yamabe solitons and gradient Einstein solitons respectively.

A Differential Inequality for the Solution of Monge Ampère Equation on Riemannian Manifold

In three-dimensional Riemannian manifolds with constant curvature, the elliptic Monge amp è re equation satisfying the homogeneous Dirichlet boundary value condition is studied. Under certain conditions, an estimate related to the solution of the equation is made, and a detailed proof of differential inequality is given.

Three-dimensional Riemannian manifolds and Ricci solitons

We characterize the three-dimensional Riemannian manifolds endowed with a semi-symmetric metric ρ-connection if its Riemannian metrics are Ricci and gradient Ricci solitons, respectively. It is proved that if a three-dimensional Riemannian manifold equipped with a semi-symmetric metric ρ-connection admits a Ricci soliton, then the manifold possesses the constant sectional curvature −1 and the soliton is expanding with λ = −2. Next, we study the gradient Ricci solitons in such a manifold. Finally, we construct a non-trivial example of a three-dimensional Riemannian manifold endowed with a semi-symmetric metric ρ-connection admitting a Ricci soliton and validate our some results.

Characterization of three-dimensional Riemannian manifolds with a type of semi-symmetric metric connection admitting Yamabe soliton

We characterize the three-dimensional Riemannian manifolds equipped with a semi-symmetric metric ρ-connection under the assumption that the Riemannian metric is a Yamabe soliton. It is shown that a three-dimensional Riemannian manifold endowed with a semi-symmetric ρ-connection, whose metric is Yamabe soliton, is a manifold of constant sectional curvature −1 and the Yamabe soliton is expanding with soliton constant −6. Finally, we give an example of a three-dimensional Riemannian manifold and validate our findings.

A new calculus for the treatment of Rytov's law in the optical fiber

In the present paper, we investigate the geometric properties of the linearly polarized light wave (LPLW) and the homothetic motion of the polarization plane traveling in optical fiber in three-dimensional Riemannian manifold. We examine the behavior of the polarized plane for the conditions that the electric field ε makes a constant angle with the Frenet vectors {e1,e2,e3} of the curve related to the optical fiber that can be considered as a space curve in Riemannian 3-space. Moreover, we give the relation between the Fermi–Walker parallel transportation laws and the homothetic motion of the polarization plane in Riemannian 3-space. The key technique here we use for examining this approach is to use quaternion algebra. We give the parametric equations of the Rytov curves that are traced curves of the polarization vector ε via quaternion product and a matrix that is similar to a Hamilton operator. By means of this matrix a new motion is defined and this motion is proven to be homothetic. For this one-parameter homothetic motion, we prove some theorems about the motion of the polarization plane traveling in optical fiber in three-dimensional Riemannian manifold. Then, we obtain the characterization of the electric field and generate the electromagnetic trajectories (εM-trajectories) along the (LPLW) in the optical fiber using the variational approach. Finally, we give various examples with Maple codes to confirms the theoretical results.