arXiv research

Study geodesic and affine Killing completeness in homogeneous affine surfaces.

Study on completeness of foliations and null Killing fields in Lorentzian manifolds.

Study torsion's impact on 2D affine Killing vectors on homogeneous surfaces.

Almost Zoll affine surface found on cylinder.

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of t…

The leaf space of a Killing Riemannian foliation is a diffeological quasifold.

We prove that a bounded affine vector field on a complete Finsler manifold is a Killing vector field. This generalizes the analogous result of Hano for Riemannian manifolds. Even though our result is more general, the proof is significantly simpler.

We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields and their casual character. We also classify their Ricci, curvature and matter col…

Develops symmetric Cartan calculus linking to Patterson-Walker metric.

Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additiona…

Conditions for Riemannian manifolds to be Euclidean spheres or spaces.

Affine vector fields on pseudo-Kähler manifolds are symplectic.

New proof for homogeneous affine surfaces with torsion.

New approach classifies conformal Killing vector fields for FLRW space-time.

The paper describes and analyzes maximal Lorentzian surfaces with a Killing field, proving completeness criteria.

Study on mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.

The first half of the thesis concerns Abelian vortices and Yang-Mills (YM) theory. It is proved that the 5 types of vortices recently proposed by Manton are symmetry reductions of (A)SDYM equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integra…

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemanni…

The Killing tensor equation is a first order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolo…

New method to determine parabolic surfaces invariant under Killing fields.

Study proves existence of multiple geodesics in a specific metric space.

In this article, we study the L2-transverse conformal Killing forms on complete foliated Riemannian manifolds and prove some vanishing theorems. Also, we study the same problems on Kahler foliations with a complete bundle-like metric.

We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descript…

The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.

Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauc…

A $3$-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part,…

Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.

Given a complete, Ricci-flat 4-manifold with a Killing field, we give an estimate on the manifold's energy in terms of a certain asymptotic quantity of the Killing field. If the Killing field has no zeros and satisfies a certain asymptotic condition, the manifold is flat.

Study on completeness in affine and statistical geometry.

Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit-length Killing vector field. 2)The projection of the image of …

The paper examines conditions for a vector field to be Killing in almost Yamabe solitons.

We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if $\nabla$ is a generic connection on a princ…

We examine moduli spaces of locally homogeneous surfaces of Type~$\mathcal{B}$ with torsion where the symmetric Ricci tensor is non-degenerate. We also determine the space of affine Killing vector fields in this context.

Killing tensors on reducible spaces are reducible, except for special cases.

New complete hypersurfaces found in affine geometry.

Paper proves total curvature for convex hypersurfaces in equiaffine space.

Introduces a variational framework for indefinite Lagrangians with specific symmetries.

Characterizes symmetric Killing tensors on specific Lie groups.

Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a…

We explain a simple construction of solutions to a family of PDE's in two dimensions which includes that defining zero scalar curvature Kahler metrics, with two Killing fields, and the affine maximal equation.

Researchers solve conformal Killing forms on Kaehler manifolds.

Study of Killing spinor-valued forms and their integrability conditions.

Classifies invariant generalised Killing spinors on Lie groups.

Study characterizes 2-Killing vector fields on complex spacetimes.

We present an algebraic procedure that finds the Lie algebra of the local Killing fields of a smooth metric. In particular, we determine the number of independent local Killing fields about a given point on the manifold. Spaces of constant curvature and locally symmetric spaces are also discussed. Furthermore, we obtai…

If the Killing vector field in a Riemannian manifold is the gradient of a smooth real valued function, then it is called Killing potential. In this paper we have deduced a necessary condition for the existence of Killing potential in a complete Riemannian manifold. Yau proved the Liouville theorem of harmonic function …

In this paper, we completely classify the magnetic curves (also N-magnetic curves with constant curvature) in a Galilean 3-space associated to a Killing vector field.

Odd-dimensional Riemannian manifolds admit pure spin-c Killing spinors if and only if they are α-Sasakian.