New equations connect unit Killing vectors to initial data.
problem Characterizing initial data for Einstein vacuum with unit Killing vectors.
method Developed new equations (uKID) by eliminating scaling and using propagation identity.
result Found equations that are finite type and characterize unit normalized Killing vectors.
The paper explores conditions for solving the Killing equation in curved spaces and spacetimes.
problem Integrating the Killing equation in curved spaces and spacetimes.
method Prolongation of the Killing equation using Young symmetrizers to derive integrability conditions.
result Explicit integrability conditions for the Killing equation are provided, limiting the number of solutions.
We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equatio…
Characterizes conformal Killing tensors and their Killing scales.
problem Characterizing conformal Killing tensors and their Killing scales.
method Differential prolongation using conformally invariant tractor calculus.
result Provides an invariant characterisation of Einstein Killing scales.
Study Killing equations for spinor-valued forms on pseudo-Riemannian manifolds.
problem Understanding solutions to Killing equations on pseudo-Riemannian manifolds.
method Introduced a system of partial differential Killing type equations for spinor-valued differential forms.
result Established the relationship between solutions of Killing equations on M and parallel fields on the metric cone over M. Derives a new connection for Killing tensors, preserving their solutions.
problem Generalizing Killing vector equations to higher rank tensors on manifolds.
method Develops a prolongation of the Killing tensor equation using projectively invariant tractor calculus.
result A projectively invariant connection that preserves solutions of the Killing tensor equation.
In this paper, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of A…
New spinorial field equation reveals geometric properties of Sasaki manifolds.
problem Exploring new spinorial field equations on Sasaki manifolds.
method Developed H-Killing spinors for 3-(α,δ)-Sasaki manifolds. result Obtained one-to-one correspondence between H-Killing spinors on dual pairs of Sasaki spaces. Researchers solved Einstein-Yang-Mills equations for arbitrary gauge groups.
problem Analyzing spacetime metrics and gauge fields with specific symmetries.
method Analytically integrated Einstein-Yang-Mills equations for a general gauge group.
result Solved Einstein-Yang-Mills equations for arbitrary gauge groups.
This text is dedicated to the real Killing equation on 3-dimensional Weyl manifolds. Any manifold admitting a real Killing spinor of weight 0 satisfies the conditions of a Gauduchon-Tod geometry. Conversely, any simply connected Gauduchon-Tod geometry has a 2-dimensional space of solutions of the real Killing equation …
Study geodesic and affine Killing completeness in homogeneous affine surfaces.
problem Geodesic and affine Killing completeness in homogeneous affine surfaces.
method Examined using the solution space of the quasi-Einstein equation.
result Characterized geodesic and affine Killing completeness in homogeneous affine surfaces.
Study characterizes 2-Killing vector fields on complex spacetimes.
problem Characterize 2-Killing vector fields on multiply twisted product spacetimes. method Determine nonlinear differential equations, find twisted functions, provide solutions, and construct examples.
result Completely describe 2-Killing vector fields and twisted functions on multiply twisted product spacetimes. New method solves Einstein constraint equations with non-constant mean curvature.
problem Solving Einstein constraint equations with non-constant mean curvature.
method Drift method, which compensates for greater analytic complexity.
result Method can handle metrics with conformal Killing but not true Killing vector fields.
We outline the solution of the Killing spinor equations of the heterotic supergravity. In addition, we describe the classification of all half supersymmetric solutions.
Classification of ground state solutions to critical Dirac equation on spheres.
problem Classifying ground state solutions of the critical Dirac equation.
method Exploiting conformal covariance and relating to the Yamabe equation.
result Ground state solutions are given by Killing spinors up to conformal diffeomorphisms.
We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold Mn admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics ηˉ on Mn×R such that $(M^n \ti…
Derives supergravity equations in coordinate-free notation.
problem Deriving supergravity equations in a coordinate-free manner.
method Coordinate-free notation for supergravity actions and equations.
result Existence of supersymmetries and generalized Killing spinor equations.
Study on type-D Ricci-flat metrics with Killing spinors and Killing vectors.
problem Characterizing and solving metrics with specific properties.
method Using Killing spinors and Killing vectors, rederive results via Toda field equations and axisymmetric solutions.
result Field equations linearize for certain metrics, including axisymmetric solutions.
We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Th. Friedrich (1980) and O. Hijazi (1986, 1992). The special solutions of the Einstein-Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of th…
Study of four-dimensional Lorentzian manifolds with real Killing spinors.
problem Characterizing and understanding four-dimensional Lorentzian manifolds with Killing spinors.
method Differential geometry and topology, Killing spinor equations, flow equations.
result Proves that the evolution flow defined by a real Killing spinor preserves the Hamiltonian and momentum constraints of the Einstein equation with negative curvature.
Defines Killing spinors and bosonic backgrounds in 5D supergravity.
problem Characterizing backgrounds in 5D supergravity.
method Calculates Spencer cohomology, defines Killing spinors, and imposes constraints on spinor connection curvature.
result Recover field equations of 5D supergravity and find new field equations for sp(1)-valued one-form. We consider Killing vector fields on standard static space-times and obtain equations for a vector field on a standard static space-time to be Killing. We also provide a characterization of Killing vector fields on standard static space-times with compact Riemannian parts.
New Ricci flow solutions found via T-duality.
problem Finding new solutions to Ricci flow equations.
method Introducing Ricci flow in generalized geometry and using T-duality.
result Solutions of Ricci flow and Killing spinor equations are exchanged under T-duality.
The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial dif…
Paper proves existence of ambient manifolds for null hypersurfaces solving Einstein equations.
problem Analyzing transverse expansion of metrics at null hypersurfaces.
method Covariant approach proving existence of ambient manifolds given asymptotic expansion and constraint equations.
result Existence of ambient manifolds solving Einstein equations to infinite order at null hypersurfaces.
We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…
The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.
problem Conditions for the existence of a third rank Killing tensor field on a 2D Riemannian torus.
method Analyzes the metric of the torus and uses Fourier coefficients to derive conditions for the function λ.
result Equations relating Fourier coefficients of the function λ determine the existence of a third rank Killing tensor field.
New proof shows all conformal vector fields on complex hyperbolic space are Killing.
problem Proving all conformal vector fields on complex hyperbolic space are Killing.
method Local, analytic, and constructive approach using Lie group model and partial differential equations.
result Every conformal vector field on complex hyperbolic space is Killing.
This paper classifies Killing superalgebras for Lorentzian 4-manifolds.
problem Understanding Killing superalgebras in Lorentzian 4-manifolds.
method Reinterpreting as filtered deformations of subalgebras, computing Spencer cohomology, identifying Killing spinors.
result Maximally supersymmetric backgrounds are classified based on their Killing superalgebras.
Spin geometry reviewed with applications in physics.
problem None explicitly stated, but related to spin geometry and its applications.
method Definitions of spin geometry, Clifford algebras, and spinors; discussion of differential operators, twistor and Killing spinors; holonomy classification; construction of symmetry operators and extended superalgebras.
result Construction of extended superalgebras and methods to find solutions of Seiberg-Witten equations.
Study symmetric Killing 2-tensors on manifolds, focusing on Sasakian and Euclidean spheres.
problem Characterize symmetric Killing 2-tensors on Riemannian manifolds.
method Analyze conditions on symmetric Killing 2-tensors, focusing on Sasakian and Euclidean spheres.
result Recover characterization of spheres using functions satisfying a differential equation.
We prove that a compact quaternionic-Kähler manifold of dimension 4n≥8 admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionic-Kähler structure.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows M carrying non-trivial solutions to that equation in case M is a local Riemannian product, a Sasakian manifold or 3-dimensional.
Researchers solve conformal Killing forms on Kaehler manifolds.
problem Classifying conformal Killing forms on compact Kaehler manifolds.
method Explicit determination of conformal Killing forms in middle degree.
result First examples of conformal Killing forms not from Hamiltonian 2-forms.
Study the algebraic structure of Killing superalgebras in 11D supergravity.
problem Classify highly supersymmetric backgrounds.
method Mapped classification problem to filtered deformations of graded subalgebras of Poincaré superalgebra.
result Reconstructing backgrounds from Killing superalgebras and relating to field equations.
Study of Killing spinor-valued forms and their integrability conditions.
problem Understanding Killing spinor-valued forms and their properties.
method Detailed treatment of prolongation and integrability conditions, relating to curvature of the manifold.
result New solutions found that are not from tensor products of Killing spinors and Killing-Yano forms.
Harmonic spinors linked to twistor spinors via potential forms and conformal Killing-Yano forms.
problem Connecting harmonic spinors and twistor spinors using mathematical operators.
method Constructing symmetry operators from conformal Killing-Yano forms and finding transformation operators between twistor and harmonic spinors in terms of potential forms.
result Operators transforming gauged twistor spinors to gauged harmonic spinors and algebraic conditions for Seiberg-Witten equations solutions.
Valence two Killing tensors in the Euclidean and Minkowski planes are classified under the action of the group which preserves the type of the corresponding Killing web. The classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebr…
Study on hypersurfaces in Einstein manifolds using Killing spinors.
problem Characterizing hypersurfaces in Einstein manifolds.
method Describes PDEs for induced spinors and proves embedding results.
result Embedding results for real analytic pseudo-Riemannian manifolds.
We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are c…
New framework for generalized Killing spinors on manifolds.
problem Formulating generalized Killing spinor equations as polyform systems.
method Developing a new framework using polyforms and algebraic relations in the Kähler-Atiyah bundle.
result Characterization of real spinor squaring map as a real algebraic variety.
We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifo…
Solves numerical computation of Killing and conformal Killing vector fields on compact Riemannian manifolds.
problem Overdetermined systems of PDE make numerical computation difficult.
method Reduces to symmetric eigenvalue problem solved by finite element techniques.
result Valid in any dimension and for arbitrary compact Riemannian manifolds.
The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.
problem Classifying quasi-Einstein structures and understanding their properties.
method Analyzing quasi-Einstein equations and exploring their connections to Hitchin's equations.
result A class of quasi-Einstein structures on closed manifolds must admit a Killing vector field.
Researchers derive symmetry operators from twistor spinors in curved spacetime.
problem Deriving symmetry operators for gauged twistor spinors in curved backgrounds.
method Using gauged twistor spinors and conformal Killing-Yano forms, symmetry operators are constructed.
result Symmetry operators can be obtained from ordinary twistor spinors in constant curvature backgrounds.
New methods show quasinormal modes can be defined using various stationary Killing vectors.
problem Proving asymptotic expansions for wave equations in Kerr-de Sitter spacetimes.
method New definition of quasinormal modes using different stationary Killing vectors.
result Horizon Killing vector fields work for analysis, simplifying the problem.
We study the spinorial Killing equation of supergravity involving a torsion 3-form $\T$ as well as a flux 4-form $\F$. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel $\G_2$-geometries and on the homogeneous Aloff-Wallach space. The constraint $\F \c…