Study on null hypersurfaces in complex contact manifolds.
problem Characterizing null hypersurfaces in indefinite complex contact manifolds.
method Proved classification results for various null hypersurfaces and characterized the ambient space.
result The ambient complex contact manifold must have constant GH-sectional curvature of −3 for certain null hypersurfaces. Study on null submanifolds in indefinite complex contact geometry.
problem Geometry of null submanifolds in indefinite complex contact manifolds.
method Analysis of quaternion null submanifolds and distributions on screen submanifolds.
result Quaternion null submanifolds are always totally geodesic.
Study on special null submanifolds in indefinite Sasakian manifolds.
problem Characterizing null submanifolds in indefinite Sasakian manifolds.
method Proving properties of screen conformal null submanifolds and defining a new class.
result Existence of contact screen conformal r-null submanifolds in indefinite Sasakian space forms. Study on lightlike submanifolds in statistical manifold geometry.
problem Characterizing contact CR and SCR-lightlike submanifolds.
method Developed characterization theorems on integrability and geodesicity.
result Obtained results on geometry of contact CR and SCR-lightlike submanifolds.
Study higher order mean curvatures in SAC half-lightlike submanifolds.
problem Exploring new geometric properties of SAC submanifolds.
method Introduced and used higher order mean curvatures; derived integration formulae.
result Generalized known results and derived new integration formulae.
New types of null hypersurfaces found in Sasakian space-forms.
problem Identifying new structures in Sasakian space-forms.
method Defined and analyzed contact screen conformal and umbilic null hypersurfaces.
result Proved these hypersurfaces exist in Sasakian space forms with specific curvature.
Study lightlike hypersurfaces in a specific type of manifold.
problem Characterize lightlike hypersurfaces in indefinite almost contact metric-manifolds.
method Analyze the position of the structure vector field and classify hypersurfaces into two types.
result Prove there are only two types of lightlike hypersurfaces: ascreen and inascreen.
This paper classifies holonomy groups of K-contact sub-pseudo-Riemannian manifolds.
problem The problem of subspace degeneracy in indefinite signature metrics.
method Adapted for metrics of indefinite signature, bypassing subspace degeneracy.
result Horizontal holonomy group either coincides with the adapted holonomy group or acts as its normal subgroup of codimension one.
In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if M=N⊥×fNT is a warped CR-submanifold such that N⊥ is φ?-anti-invariant and NT is φ?-invariant, then M is a CR-product. We…
Study geometric properties of SGL submanifolds in a specific manifold.
problem Analyzing geometric characteristics of SGL submanifolds.
method Examines integrability conditions and parallelism properties of distributions.
result Provides insights into geometric behavior of SGL submanifolds.
We study the indefinite metric G in the contact phase space (P,θ) of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of P - constitutive surfaces of different homogeneous thermodynamical syste…
This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric and almost hypercomplex manifolds with Hermitian and anti-Hermitian metric.
3-Sasaki structures linked to projective geometry.
problem Understanding 3-Sasaki structures via projective geometry.
method Establishing a connection between 3-Sasaki structures and projective structures with specific holonomy reductions.
result 3-Sasaki structures are described as projective structures with a particular holonomy reduction to the unitary quaternionic group.
Study on lightlike geometry in indefinite Sasakian statistical manifolds.
problem Exploring lightlike hypersurfaces and their properties in indefinite Sasakian statistical manifolds.
method Introducing indefinite Sasakian statistical manifolds and analyzing lightlike hypersurfaces with respect to dual connections.
result An invariant lightlike submanifold of an indefinite Sasakian statistical manifold is itself an indefinite Sasakian statistical manifold.
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
Study new Hopf real hypersurfaces in indefinite complex projective space.
problem Problems posed by H.~Anciaux and K.~Panagiotidou on non-degenerate real hypersurfaces.
method Changed point of view, constructed new families, obtained rigidity results.
result Classified η-umbilical real hypersurfaces and characterized Killing Reeb vector field. Study lightlike submanifolds in indefinite statistical manifolds, finding conditions and curvature expressions.
problem Characterize lightlike submanifolds in indefinite statistical manifolds.
method Analyze conditions for lightlike submanifolds to be lightlike statistical submanifolds, derive statistical sectional curvature, and investigate induced statistical Ricci tensor symmetry.
result Conditions for lightlike submanifolds to be lightlike statistical submanifolds and expressions for statistical sectional curvature and induced Ricci tensor symmetry.
We present a compared analysis of some properties of indefinite almost S-manifolds and indefinite S-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and φ-sectional curvature of indefinite almost $\…
Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.
problem Characterizing curvature symmetries in lightlike hypersurfaces of indefinite Sasakian manifolds.
method Analyzing properties of lightlike hypersurfaces in indefinite Sasakian manifolds.
result Lightlike hypersurfaces are not locally symmetric, semi-symmetric, or semi-parallel.
An η-Einstein condition is introduced in the context of indefinite g.f.f-manifolds, and a few Schur-type lemmas for indefinite S-manifolds are provided.
Study on SASI-lightlike submanifolds in indefinite Kaehler manifolds.
problem Characterizing and understanding SASI-lightlike submanifolds in indefinite Kaehler manifolds.
method Introduced and analyzed SASI-lightlike submanifolds, derived conditions for metric connection, and investigated integrability of distributions.
result Found necessary and sufficient conditions for the induced connection to be a metric connection on SASI-lightlike submanifolds.
Study new CR-lightlike submanifolds in indefinite nearly Sasakian manifolds.
problem Characterizing new CR-lightlike submanifolds in indefinite nearly Sasakian manifolds.
method Introduce and study Quasi Generalized Cauchy-Riemann (QGCR) lightlike submanifolds.
result Derive necessary and sufficient conditions for integrability of distributions.
Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.
problem Existence of lightlike hypersurfaces in indefinite Sasakian manifolds.
method Proved non-existence through properties of second fundamental forms and induced structural tensors.
result No lightlike hypersurfaces can have parallel or recurrent second fundamental forms or induced structural tensors.
Construct broken Lefschetz fibrations from handle decompositions.
problem Constructing explicit broken Lefschetz fibrations.
method Using handle decompositions of 4-manifolds to construct BLFs explicitly.
result The approach works in the case when X is the double of a 4-manifold with boundary. We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
We study the basic geometric properties of an indefinite locally conformal Kaehler manifold.
We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reve…
New indefinite false theta functions match homological blocks for a specific 3-manifold.
problem Constructing homological blocks for certain 3-manifolds.
method Developing indefinite false theta functions and proving their equivalence to homological blocks.
result Indefinite false theta functions match homological blocks for the Poincaré homology sphere.
Improved indefinite Kasparov modules for non-elliptic operators.
problem Generalizing unbounded Kasparov modules for non-symmetric operators.
method New theorem on self-adjointness and regularity of weakly anticommuting operators.
result Equivalence between indefinite Kasparov modules and pairs of Kasparov modules.
Some curvature properties of Kahler manifolds of indefinite metrics are studied. Analogues of a Kulkarni's theorem are proved for such manifolds.
Study of spheres and circles on a manifold with a specific metric structure.
problem Understanding geometric objects on a manifold with a skew-circulant structure.
method Analyzing hyper-spheres, spheres, and circles in a tangent space of a 4D manifold with a skew-circulant tensor structure.
result Characterization of geometric objects under an indefinite metric.
Construct algorithms to simplify indefinite fibrations on 4-manifolds.
problem Simplify topology of indefinite fibrations on 4-manifolds.
method Explicit algorithms using sequences of moves to modify fibrations in families.
result Existence of simplified broken Lefschetz fibrations and Morse 2-functions on general 4-manifolds.
The paper bounds the genus of surfaces in four-manifolds with indefinite forms.
problem Bounding the genus of surfaces in four-manifolds with indefinite intersection forms.
method Using adjunction inequalities and the 10/8 + 4 theorem, the paper provides methods to bound the genus of surfaces.
result The set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.
A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions …
In a metric g.f.f-manifold we study lightlike hypersurfaces M tangent to the characteristic vector fields, and owing to the presence of the f-structure, we determine some decompositions of TM and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the …
Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.
problem Determining the minimal number of homogeneous geodesics in Finsler manifolds with indefinite Killing form.
method Analyzing examples of Lie groups with invariant Finsler metrics and presenting new examples.
result Homogeneous Finsler manifolds with indefinite Killing form admit at least four homogeneous geodesics.
Introduces new lightlike submanifolds in indefinite Kaehler manifolds.
problem Characterizing and understanding new types of lightlike submanifolds.
method Definition and analysis of Screen Transversal Cauchy Riemann (STCR) lightlike submanifolds.
result Characterizes STCR lightlike submanifolds as umbrella of other types.
Introduces a variational framework for indefinite Lagrangians with specific symmetries.
problem Handling indefinite Lagrangians with complex symmetries.
method Develops a variational setting for an indefinite Lagrangian with a specific Noether charge.
result Validates the existence of a variational setting for a broad class of Lagrangians.
Study finds equations for spheres and circles on a specific manifold.
problem Equations for spheres and circles on a manifold with circulant structures.
method Analyzes a 3D manifold with Riemannian and additional indefinite metrics.
result Equations for spheres and circles defined with respect to the associated metric.
Conditions, related to Kulkarni's equivalence problem are considered for indefinite Riemannian and Kaehlerian manifolds. Corresponding theorems are obtained for the values of the Ricci tensor on isotropic vectors as well as for the values of the curvature tensor on degenerate holomorphic 2-planes.
The paper finds a new lower bound on the genus of surfaces in indefinite 4-manifolds.
problem Finding a new lower bound on the genus of surfaces in indefinite 4-manifolds.
method Proves a new lower bound on the genus of a properly embedded surface in X∖B4 representing a given homology class and with boundary a quasipositive knot K⊂S3. result The minimal genus of such a surface is equal to the slice genus of K in the null-homologous case. Study adiabatic limits of G_2-manifold fibrations.
problem Understanding the adiabatic limit of co-associative fibrations.
method Propose a maximal submanifold in a space of indefinite signature.
result Global versions of constructions set up for adiabatic limit.
Authors study a new type of submanifolds in a specific type of manifold.
problem Exploring a new type of submanifolds in indefinite nearly μ-Sasakian manifold. method Prove existence theorem and classification theorems for minimal and co-screen quasi generalized CR-lightlike submanifolds.
result Existence and classification theorems for specific submanifolds.
Study characterizes geometric properties of QGCR-null submanifolds in cosymplectic manifolds.
problem Characterizing geometric properties of QGCR-null submanifolds in cosymplectic manifolds.
method Characterization through totally umbilical and irrotational properties, and discussion of geodesity conditions.
result Characterizes geometric properties of ascreen QGCR-null submanifolds.
This thesis extends noncommutative geometry to semi-Riemannian manifolds and applies it to gauge theories.
problem Applying noncommutative geometry to Lorentzian manifolds for particle physics.
method Generalizing spectral triples to semi-Riemannian manifolds, constructing noncommutative gauge theories.
result Recovering the Standard Model using noncommutative geometry.
This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local stru…
The study extends Hano's theorem to semi-Riemannian product manifolds with specific conditions.
problem Extending Hano's theorem to manifolds with indefinite metrics.
method Generalization of Hano's theorem to semi-Riemannian product manifolds with specific conditions.
result The assumption on the factors is necessary for the generalization.
In this paper we study the 3-dimensional (ε)-para Sasakian manifolds. We obtain an necessary and sufficient condition for an (ε)-para Sasakian 3 -manifold to be an indefinite space form. We show that a Ricci-semi-symmetric (ε)-para Sasakian 3 -manifold is an indefinite space form…