Geodesic Vector Field Research Articles

Slant Curves in Contact Lorentzian Manifolds with CR Structures

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 ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

Rigged null hypersurfaces in almost paracontact metric manifolds

Given an almost paracontact metric manifold $(\overline {M}, \overline{ \phi},\overline {\eta}, \xi, \overline {g})$, we study lightlike hypersurfaces $M$ of the semi-Riemannian manifold $(\overline {M},\overline{g})$ transversal to the structure vector field $\xi$. The latter is then a rigging for $M$ and defines a null section $E$ of the radical distribution of $M$ and a screen distribution which turns out to be always semi-invariant. We show that leaves of an integrable screen distribution in such hypersurfaces $M$ are almost paracontact metric manifolds too. When the ambient space $(\overline {M}, \overline{ \phi},\overline {\eta}, \xi, \overline {g})$ is para-Sasakian, we show that the screen distribution cannot be conformal and that $E$ is a geodesic vector field in $(M, \nabla),$ where $\nabla$ is the connection induced on $M$ by Levi-Civita connection of $\overline {g}$ and the local null rigging $N=\xi-\frac{1}{2}E$. We also find necessary and sufficient conditions for the leaves of the screen distribution to be para-Sasakian too and finally we investigate integrability conditions for some additional distributions induced on $M$ by the structure $(\overline {M}, \overline{ \phi},\overline {\eta}, \xi, \overline {g})$.

Some remarks on Yamabe solitons

The evolution of some geometric quantities on a compact Riemannian manifold [Formula: see text] whose metric is Yamabe soliton is discussed. Using these quantities, lower bound on the soliton constant is obtained. We discuss about commutator of soliton vector fields. Also, the condition of soliton vector field to be a geodesic vector field is obtained.

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.

Stability and Instability of the Sub-extremal Reissner\u2013Nordstr\xf6m Black Hole Interior for the Einstein\u2013Maxwell\u2013Klein\u2013Gordon Equations in Spherical Symmetry

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting:Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry.Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry.This instability of the black hole interior can also be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data.

Some results on almost Ricci solitons and geodesic vector fields

We show that a compact almost Ricci soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Next we show that an almost Ricci soliton reduces to Ricci soliton if and only if the associated vector field is geodesic. Finally, we prove that a contact metric manifold is K-contact if and only if its Reeb vector field is geodesic.

On pseudo-Finsler manifolds of scalar flag curvature

Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated to the non-degenerate, 2-positively homogeneous Lagrangian L . In this paper we prove that ( M , L ) is of scalar flag curvature k if and only if the equation L ξ g + k λ L ξ g ˆ = 0 holds on Γ ( I λ M ) , the Lie algebra of tangent vector fields to the λ -indicatrix bundle I λ M , where g and g ˆ are pseudo-Riemannian metrics on the vertical and respectively on the horizontal subbundle. Also, we prove that any pseudo-Finsler manifold is of scalar flag curvature at any point of the light cone.

Approach of background metric expansion to a new metric ansatz for gauged and ungauged Kaluza-Klein supergravity black holes

In a previous paper [S.Q. Wu, Phys. Rev. D 83, 121502(R) (2011)], a new kind of metric ansatz was found to fairly describe all already known black hole solutions in the ungauged Kaluza-Klein (KK) supergravity theories. That metric ansatz somewhat resembles to the famous Kerr-Schild (KS) form, but it is different from the KS one in two distinct aspects. That is, apart from a global conformal factor, the metric ansatz can be written as a vacuum background spacetime plus a "perturbation" modification term, the latter of which is associated with a timelike geodesic vector field rather than a null geodesic congruence in the usual KS ansatz. In this paper, we shall study this novel metric ansatz in detail, aiming at achieving some inspiration as to the construction of rotating charged AdS black holes with multiple charges in other gauged supergravity theories. In order to investigate the metric properties of the general KK-AdS solutions, in this paper we devise a new effective method, dubbed the background metric expansion method, which can be thought of as a generalization of the perturbation expansion method, to deal with the Lagrangian and all equations of motion. In addition to two previously known conditions, namely the timelike and geodesic properties of the vector, we get the three additional constraints via contracting the Maxwell and Einstein equations once or twice with this timelike geodesic vector. In particular, we find that these are a simple set of sufficient conditions to determine the vector and the dilaton scalar around the background metric, which is helpful in obtaining new exact solutions. With these five simplified equations in hand, we rederive the general rotating charged KK-(A)dS black hole solutions with spherical horizon topology and obtain new solutions with planar topology in all dimensions.

Homogeneous geodesics of non-reductive homogeneous pseudo-Riemannian 4-manifolds

After obtaining an explicit description in coordinates of invariant metrics on four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds, for each of these spaces we completely describe Killing and geodesic vector fields and homogeneous geodesics through a point. In particular, this allows us to determine explicit examples of four-dimensional non-reductive pseudo-Riemannian g.o. spaces. Finally, we determine the possible holonomy Lie algebras of the invariant metrics.

Generalized Killing spinors and Lagrangian graphs

We study generalized Killing spinors on the standard sphere S3, which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold S3×S3 and to great circle flows on S3. Using our methods we generalize a well known result of Gluck and Gu [5] concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of S3×S3 has at least three connected components.

Geodesics on a supermanifold and projective equivalence of superconnections

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field associated with the connection. Our (super) geodesics possess the same properties as in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass. Moreover, using our definition, we are able to establish Weyl’s characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) 1-form.

Null scrolls as fluctuating surfaces: a new simple way to construct extrinsic string solutions

We exhibit a surprising phenomenon that happens in the conformal Lorentz Minkowski three space, which has no counterpart in a Riemannian setting. Whenever a curve, no matter its causal character, propagates transversely through a conformal geodesic null vector field, it is generating the worldsheet of an extrinsic Polyakov string solution. Furthermore, the Polyakov extrinsic energy of these solutions only depend on the world- sheet topology and it can be computed not only intrinsically, but also holographically by measuring the hyperbolic angles in the boundary corners. This geometric approach, to provide extrinsic string solutions, can be considered as an alternative to the Pohlmeyer reduced mechanism. Then, we describe how to translate these solutions to the language of Pohlmeyer theory. We will also show that any curve in the conformal boundary of the anti de Sitter 3-space can be viewed as a piece of the generalized Wilson loops associated with an extrinsic string solution obtained by this geometric mechanism.

Geodesic patterns

Geodesic curves in surfaces are not only minimizers of distance, but they are also the curves of zero geodesic (sideways) curvature. It turns out that this property makes patterns of geodesics the basic geometric entity when dealing with the cladding of a freeform surface with wooden panels which do not bend sideways. Likewise a geodesic is the favored shape of timber support elements in freeform architecture, for reasons of manufacturing and statics. Both problem areas are fundamental in freeform architecture, but so far only experimental solutions have been available. This paper provides a systematic treatment and shows how to design geodesic patterns in different ways: The evolution of geodesic curves is good for local studies and simple patterns; the level set formulation can deal with the global layout of multiple patterns of geodesics; finally geodesic vector fields allow us to interactively model geodesic patterns and perform surface segmentation into panelizable parts.

Gauduchon-Tod structures,Simholonomy and De Sitter supergravity

Solutions of five-dimensional De Sitter supergravity admitting Killing spinors are considered, using spinorial geometry techniques. It is shown that the solutions are defined in terms of a one parameter family of 3-dimensional constrained Einstein-Weyl spaces called Gauduchon-Tod structures. They admit a geodesic, expansion-free, twist-free and shear-free null vector field and therefore are a particular type of Kundt geometry. When the Gauduchon-Tod structure reduces to the 3-sphere, the null vector becomes recurrent, and therefore the holonomy is contained in Sim(3), the maximal proper subgroup of the Lorentz group SO(4,1). For these geometries, all scalar invariants built from the curvature are constant. Explicit examples are discussed.

SLANT CURVES IN CONTACT PSEUDO-HERMITIAN 3-MANIFOLDS

AbstractBy using the pseudo-Hermitian connection (or Tanaka–Webster connection) $\widehat \nabla $, we construct the parametric equations of Legendre pseudo-Hermitian circles (whose $\widehat \nabla $-geodesic curvature $\widehat \kappa $ is constant and $\widehat \nabla $-geodesic torsion $\widehat \tau $ is zero) in S3. In fact, it is realized as a Legendre curve satisfying the $\widehat \nabla $-Jacobi equation for the $\widehat \nabla $-geodesic vector field along it.

Conformal killing vector fields on spacetime solutions of Einstein's equations and initial data

Working within the ADM formalism we have established results characterizing a conformal Killing vector (CKV) field on a spacetime solution of Einstein's field equations and the initial data on a spacelike hypersurface. We have considered the cases (i) the CKV is tangential to spacelike slices, (ii) the CKV is closed and the initial hypersurface is totally umbilical and has constant mean curvature (CMC), and (iii) spacetime is a perfect fluid, has a non-vanishing CKV transverse to a compact CMC hypersurface orthogonal to the 4-velocity. Finally, we show that a non-null CKV that is also a geodesic vector field is either homothetic or locally closed.

On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations

We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a totally umbilic hyperfoliation. A corresponding example shows the non-triviality of this condition. In the 2-dimensional case, we give a complete description of Riemannian manifolds admitting a geodesic unit vector field with totally geodesic property.

A totally geodesic property of Hopf vector fields

We prove that the Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As an application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field.

Harmonic and minimal vector fields on tangent and unit tangent bundles

We show that the geodesic flow vector field on the unit tangent sphere bundle of a two-point homogeneous space is both minimal and harmonic and determines a harmonic map. For a complex space form, we exhibit additional unit vector fields on the unit tangent sphere bundle with those properties. We find the same results for the corresponding unit vector fields on the pointed tangent bundle. Moreover, the unit normal to the sphere bundles in the pointed tangent bundle of any Riemannian manifold always enjoys those properties.

The algebra of generalized jacobifields

We study the structure of those vector fields on the tangent bundle of an arbitrary smooth manifold which commute with the geodesic vector field defined by an affine connection. The study is restricted to polylinear fields generated by a pair of symmetric pseudotensor fields of type (k, 1) and (k+1,1), k≥0, defined on the manifold. We establish an isomorphism between the space of infinitesimal automorphisms of fixed type and the space ℌk of the solutions of a partial differential equation generalizing the Jacobi equation for the infinitesimal automorphisms of the connection. It is shown that the spaces ℌk are finite-dimensional and form a graduated Lie algebra ℌ=⊕ k=0 ∞ ℌk. These algebras are classified in the case of one-dimensional manifolds. It is proved that if the geodesic vector field is complete, then so are the automorphisms corresponding to covariant constant fields of type (1, 1). Bibliography: 5 titles.