New conditions found for Riemannian foliations with specific conformal fields.
problem Conditions for Riemannian foliations with transversal conformal fields.
method Analyzes conditions for transversal isometry of foliations.
result Found conditions for F to be isometric to the sphere. Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field
Let M be a closed manifold which admits a foliation structure F of codimension q≥2 and a bundle-like metric g0. Let [g0]B be the space of bundle-like metrics which differ from g0 only along the horizontal directions by a multiple of a positive basic function. Assume Y is a transverse con…
Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.
problem Investigate foliations transverse to closed conformal vector fields on Riemannian manifolds.
method Analyze conditions for totally geodesic leaves and geometric constraints on foliations.
result Characterize totally geodesic foliations and classify minimal and constant mean curvature foliations.
String backgrounds yield simplified Hull-Strominger system solutions.
problem Solving the simplified Hull-Strominger system in various geometries.
method Variational argument using string action, gradient Ricci solitons, and symmetry reduction.
result Canonical symmetry and transverse geometry properties derived.
On a closed, connected Riemannian manifold with a Kähler foliation of codimension q=2m, any transverse Killing r (≥2)-form is parallel (S. D. Jung and M. J. Jung [\ref{JJ2}], Bull. Korean Math. Soc. 49 (2012)). In this paper, we study transverse conformal Killing forms on Kähler foliations and prove that if th…
We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension q>1 and a bundle-like metric. Then (M,F) is transversally isometric to the q-sphere of radius 1/c in (q+1)-dimensional Euclidean space endowed with the action of a discrete subgroup of th…
Study transverse metric expansion on null hypersurfaces, proving uniqueness for Killing horizons.
problem Analyzing transverse expansion of metric on null hypersurfaces.
method Covariant approach, general geometric identities, generalized symmetry generators.
result Transverse expansion of spacetime metric uniquely determined at non-degenerate Killing horizons.
We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with C-positive normal curvature, if there is a closed basic 1-form φ such that ΔBφ=qCφ, then the foliation is transversally isometric to the quotient of a q-sphere.
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.
Study on biharmonic maps between conformally compact manifolds, proving non-existence under certain conditions.
problem Analyzing biharmonic maps on conformally compact manifolds.
method Investigating simple b-maps, focusing on non-existence results for biharmonic maps. result Non-existence of biharmonic maps under specific conditions, leading to implications for minimal surfaces.
In this paper we generalize the basic Lichnerowicz cohomology on transversally locally conformally Kählerian foliations and we study its relation with basic Bott-Chern cohomology and 0--th basic Dolbeault cohomology with values in the associated foliated weight bundle.
Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the conformal geometry of such manifolds.
Study curvature lines of a vector field on surfaces.
problem Behavior of curvature lines at umbilical points.
method Analyzes transversal eqüiaffine vector fields on surfaces.
result Behavior of curvature lines at isolated umbilical points.
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.
problem Determining an unknown function in a semilinear elliptic equation on complex manifolds.
method Analyzing higher order linearizations and interactions of Gaussian quasimode solutions.
result An unknown smooth function can be uniquely determined from the Dirichlet-to-Neumann map.
The paper analyzes null infinity's geometry without restrictions.
problem Understanding null infinity's geometry without constraints.
method Coordinate-free approach, treating conformal factor as dynamical.
result Isometric spacetimes with identical free data at null infinity.
Geodesic vector fields on flat 3-manifolds are related to contact structures.
problem Understanding geodesic vector fields on flat 3-manifolds.
method Analyzing geodesic and Reeb vector fields on flat 3-manifolds.
result Geodesic vector fields on closed flat 3-manifolds are Reeb vector fields of contact forms.
Study simplifies classification of foliations with specific geometric structures.
problem Classifying foliations with transverse similarity structures.
method Conceptual approach and proof of important results.
result Holonomy classification of Weyl structures on compact manifolds.
We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …
We construct rigid supersymmetric gauge theories on Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of a six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals. …
In the paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.
We consider vector fields on knot/link complements in S3 which are transverse to the fibres of a fibration of the complement over a circle. We prove that a large class of fibred knots/links, including all non-torus fibred 2-bridge knots, has the following property: any vector field transverse to the fibres of the fi…
The paper proves generic transversality and regularity for minimal submanifolds and area-minimizing currents.
problem Transversality and regularity of minimal submanifolds and area-minimizing currents.
method Proves transversality and regularity for generic metrics using Baire category and minimization properties.
result Generic metrics ensure transversality and regularity for minimal submanifolds and area-minimizing currents.
Let (M, g) be a simple, real analytic, Riemannian manifold with boundary and of dimension n>=3. In this work, we prove a support theorem for the transverse ray transform of tensor fields of rank 2 defined over such manifolds. More specifically, given a symmetric tensor field f of rank 2, we show that if the transverse …
Generalizing local Gromov-Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and…
Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.
problem Realizing geometric Langlands correspondences for Higgs bundles and connections.
method Using transversal Higgs bundles and holomorphic connections, induced divisors and parameters.
result Evidence suggests generic realization of Dolbeault geometric Langlands correspondence.
It is well-known that if ξ is a smooth vector field on a given Riemannian manifold Mn then ξ naturally defines a submanifold ξ(Mn) transverse to the fibers of the tangent bundle TMn with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We sh…
Classifies vector fields in the kernel of a 1-form, up to equivalence.
problem Classifying vector fields in the kernel of a 1-form.
method Equivalence relation, local models, transversal unfoldings.
result Provides a list of local models and transversal unfoldings for vector fields.
The paper bounds eigenvalues and integrals of eigenfunctions on hyperbolic manifolds.
problem Eigenvalues and integrals of eigenfunctions on compact hyperbolic manifolds.
method Spectral decompositions and consistency conditions derived from quadruple overlap integrals.
result Upper bounds on Laplacian eigenvalues and triple overlap integrals.
Consider a smooth manifold M with a smooth cometric g∗ which changes the bilineal type by transverse way, on a hypersurface D∞. Suppose that the radical annihilator hyperplane is tangent to D∞. We examine the geometry of the (g∗-dual) covariant metric g on M− D∞, prov…
Study finds critical points of volume functionals on Sasaki manifolds.
problem Finding Kähler-Einstein metrics on Sasaki manifolds.
method Revisited moment polytopes, applied to volume minimization.
result Transverse coupled Kähler-Einstein metrics found as critical points.
Classifies singularities of smooth vector fields on the line.
problem Classifying singularities of smooth vector fields on the line.
method Local classification with respect to C1-conjugacy, including normal forms and unfoldings. result Complete description of the 1-d case achieved.
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
Equivalence found between certain Kahler and Sasaki metrics.
problem Understanding relationships between Kahler and Sasaki metrics.
method Establishing an equivalence between conformally Einstein-Maxwell Kahler 4-manifolds and extremal Kahler 4-manifolds with non-vanishing scalar curvature.
result New existence and non-existence results for extremal Sasaki metrics.
We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…
Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki manifolds.
The paper establishes a connection between force-free fields and conformally geodesic fields.
problem Understanding the relationship between force-free fields and conformally geodesic fields.
method Developed an equivalence between force-free fields and conformally geodesic fields, generalized to arbitrary dimensions.
result Established that stationary points of hierarchies of L2 and L1-optimization problems are related by a conformal change of metric. We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a 3-manifold M that are transverse to a nowhere-zero vector field V up to the corresponding isotopy relation. Such knots are called …
Paper proves embedding theorem for conformally compact manifolds.
problem Embedding conformally compact manifolds into hyperbolic spaces.
method Proves analogous Nash Embedding Theorem for conformally compact manifolds.
result Conformally compact manifolds can be isometrically embedded into hyperbolic spaces.
Study of instantons on Sasakian 7-manifolds using gauge fields.
problem Understanding instantons on Sasakian manifolds.
method Fredholm theory, cohomological conditions, index of a transverse elliptic operator.
result Moduli space of selfdual contact instantons is Kähler.
New approach classifies conformal Killing vector fields for FLRW space-time.
problem Classifying conformal Killing vector fields for FLRW space-time.
method Introduced new perspective on conformal Killing vector fields for FLRW space-time, considering three cases for the conformal factor.
result Nine conformal vector fields on FLRW, six of which are Killing and the rest non-Killing.
Study classifies Einstein 4-manifolds with specific curvature properties.
problem Classifying Einstein 4-manifolds with specific curvature properties.
method Classification based on self-dual Weyl curvature and harmonic forms.
result Similar results obtained for non-negative curvature, differing from positive curvature cases.
Expanding on supersymmetric Yang-Mills theory, this work focuses on cohomological field theory aspects.
problem Exploring the mathematical aspects of supersymmetric Yang-Mills theory on 4D manifolds.
method Constructing a cohomological field theory using the Atiyah-Jeffrey construction and demonstrating the transversally elliptic complex.
result The deformation theory of flipping instantons is controlled by a transversally elliptic complex, crucial for non-degeneracy and calculability.
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.
problem Characterizing conformal vector fields on lcK manifolds.
method Analyzing properties of conformal vector fields on compact lcK manifolds.
result Conformal vector fields on compact lcK manifolds are either Killing or holomorphic.
The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.
problem Proving triviality of modified conformal vector fields on Riemannian manifolds.
method Analyzing properties of homothetic, conformal, and gradient vector fields on compact and non-compact manifolds.
result Established conditions under which m-modified conformal vector fields are trivial. Study of Gaussian random fields on manifolds, focusing on their differential topology.
problem Understanding the differential topology of Gaussian random fields on manifolds.
method Systematic study using Gaussian measures and weak Whitney topology, focusing on convergence in law and transversality.
result The convergence in law of Gaussian random fields is related to the convergence of their covariance structures, with important technical tools like the Thom transversality theorem.