Three-dimensional Riemannian Manifold Research Articles

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.

Loxodromes on Invariant Surfaces in Three-Manifolds

In this paper, we prove some results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, a part of which generalizes classical results about loxodromes on rotational surfaces in $${{\mathbb {R}}}^3$$. In particular, we show how to parametrize a loxodrome on an invariant surface of $${\mathbb {H}}^2\times {{\mathbb {R}}}$$ and $${\mathbb {H}}_3$$, and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.

Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D Riemannian manifold with Bishop equations

We review the geometric evolution of a linearly polarized light wave coupling into an optical fiber and the rotation of the polarization plane in a three-dimensional (3D) Riemannian manifold. The optical fiber is assumed to be a one-dimensional object imbedded in the 3D Riemannian manifold along the paper. Thus, in the 3D Riemannian manifold, we demonstrate that the evolution of a linearly polarized light wave is associated with the Berry phase or more commonly known as the geometric phase. The ordinary condition for parallel transportation is defined by the Fermi–Walker parallelism law. We define other Fermi–Walker parallel transportation laws and connect them with the famous Rytov parallel transportation law for an electric field E, which is considered as the direction of the state of the linearly polarized light wave in the optical fiber in the 3D Riemannian manifold. Later, we define a special class of magnetic curves called by Bishop electromagnetic curves (BEM-curves), which are generated by the electric field E along the linearly polarized monochromatic light wave propagating in the optical fiber. In this way, not only we define a special class of linearly polarized point-particles corresponding to BEM-curves of the electromagnetic field along with the optical fiber in the 3D Riemannian manifold, but we also calculate, both numerically and analytically, the electromagnetic force, Poynting vector, energy-exchanges rate, optical angular and linear momentum, and optical magnetic-torque experienced by the linearly polarizable point-particles along with the optical fiber in the 3D Riemannian manifold.

Asymptotically sectional-hyperbolic attractors

The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Furthermore, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.

On the local isometric embedding of trapped surfaces into three-dimensional Riemannian manifolds

We study trapped surfaces from the point of view of local isometric embedding into 3D Riemannian manifolds. When a two-surface is embedded into 3D Euclidean space, the problem of finding all surfaces applicable upon it gives rise to a non-linear partial differential equation of the Monge–Ampère type, first discovered by Darboux, and later reformulated by Weingarten. Even today, this problem remains very difficult, despite some remarkable results. We find an original way of generalizing the Darboux technique, which leads to a coupled set of six non-linear partial differential equations. For the 3-manifolds occurring in Friedmann–(Lemaitre)–Robertson–Walker cosmologies, we show that the local isometric embedding of trapped surfaces into them can be proved by solving just one non-linear equation. Such an equation is here solved for the three kinds of Friedmann model associated with positive, zero, negative curvature of spatial sections, respectively.

Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

AbstractLet $(M,g)$ be a three-dimensional Riemannian manifold. The goal of the paper is to show that if $P_{0}\in M$ is a nondegenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than $32\pi $; moreover, it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the 1st multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass.

Minimal Surfaces in Three-Dimensional Riemannian Manifold Associated with a Second-Order ODE

We show that a surface corresponding to a first-order ODE is minimal in three-dimensional Riemannian manifold which is determined by the first prolongation of $${\text {d}}y/\mathrm{d}x=p(x,y)$$ if and only if $$p_{yy}=0$$ . Accordingly, any linear first-order ODE describes a minimal surface which is not necessarily totally geodesic.

A criterion for the existence of Killing vectors in 3D

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or no independent Killing vectors. We present an explicit algorithm for the computing dimension of infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curvature, but they are not scalar polynomial Weyl invariants. We compare our obstructions to the existence of Killing vectors with the known existence criteria due to Singer, Kerr and others.

On Weyl’s Embedding Problem in Riemannian Manifolds

Abstract We consider a priori estimates of Weyl’s embedding problem of $(\mathbb{S}^2, g)$ in general three-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang [18], we obtain an isometric embedding of $(\mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl’s embedding theorem in space form under the condition that $g\in C^2$ with $D^2g$ Dini continuous.

Fluids, geometry, and the onset of Navier–Stokes turbulence in three space dimensions

A theory for the evolution of a metric g driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash–Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving C1 manifold. The theory is applied to the incompressible Euler and Navier–Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier–Stokes equations.

Free boundary minimal annuli in convex three-manifolds

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds of nonnegative Ricci curvature. This includes strictly convex domains in $\mathbb{R}^3$, thereby solving an open problem of Jost.

On the number of bound states of point interactions on hyperbolic manifolds

We study the bound state problem for [Formula: see text] attractive point Dirac [Formula: see text]-interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have [Formula: see text] bound states and give an explicit criterion for it in hyperbolic manifolds [Formula: see text] and [Formula: see text]. Furthermore, we study the same spectral problem for a relativistic extension of the model on [Formula: see text] and [Formula: see text].

Geometric characterizations of asymptotic flatness and linear momentum in general relativity

In 1996, Huisken–Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Their decay assumptions were weakened by Metzger, Huang, Eichmair–Metzger, and the author at a later stage. In this work, we prove the reverse implication, i.e. any three-dimensional Riemannian manifold is asymptotically flat if it possesses a CMC- cover satisfying certain geometric curvature estimates, a uniqueness property, a weak foliation property, while each surface has weakly controlled instability. With the author's previous result that every asymptotically flat manifold possesses a CMC-foliation, we conclude that asymptotic flatness is characterized by existence of such a CMC-cover. Additionally, we use this characterization to provide a geometric (i.e. coordinate-free) definition of a (CMC-)linear momentum and prove its compatibility with the linear momentum defined by Arnowitt–Deser–Misner.

Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry

In 1996, Huisken–Yau showed that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed CMC-surfaces if it is asymptotically equal to the (spatial) Schwarzschild solution and has positive mass. Their assumptions were later weakened by Metzger, Huang, Eichmair–Metzger and others. We further generalize these existence results in dimension three by proving that it is sufficient to assume asymptotic flatness and non-vanishing mass to conclude the existence and uniqueness of the CMC-foliation and explain why this seems to be the conceptually optimal result. Furthermore, we generalize the characterization of the corresponding coordinate CMC-center of mass by the ADM-center of mass proven previously by Corvino–Wu, Huang, Eichmair–Metzger and others (under other assumptions).

Riemannian manifolds with two circulant structures

We consider a three-dimensional Riemannian manifold equipped with two circulant structures - a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector.

RESONANCES ON HEDGEHOG MANIFOLDS

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.

The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds

Let $(M,g)$ be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the $3$-sphere ${\Bbb S}^3$ with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schr\"odinger equation $i\partial_t u+\Delta u= \pm |u|^4u$, $u|_{t=0}=u_0$ is solved for small initial data $u_0$ in the energy space $H^1(M)$, which is the scaling-critical space. Further, local well-posedness for large data, as well as persistence of higher initial Sobolev regularity is obtained. This extends previous results of Burq-G\'erard-Tzvetkov to the endpoint case.