Study on complex spaces and their properties.
problem Characterizing relative properties of complex spaces.
method Analyzing indefinite Euclidean and non-flat complex spaces.
result Indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form.
We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family…
We construct invariant complex product (hyperparacomplex, indefinite quaternion) structures on the manifolds underlying the real noncompact simple Lie groups $SL(2m-1,\RR)$, SU(m,m−1) and $SL(2m-1,\CC)^\RR$. We show that on the last two series of groups some of these structures are compatible with the biinvariant Kil…
New non-flat, non-homogeneous geometries with symmetries are created.
problem Creating non-flat, non-homogeneous parabolic geometries with symmetries.
method Constructing a series of examples of parabolic geometries with specific properties.
result Examples of non-flat, non-homogeneous parabolic geometries with symmetries are successfully constructed.
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.
The paper classifies PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
problem Characterizing PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
method Analyzing the properties of hypersurfaces with at most two distinct principal curvatures.
result PMCV hypersurfaces are either minimal or locally isoparametric.
In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in Ckm. We introduce the notion of indefinite special Lagrangian submanifolds in Ckm and generalize the well-known work of Harvey-Lawson to the indefinite case.
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.
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 …
We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.
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 $\…
In this paper, we show that there exists no equifocal submanifold with non-flat section in four irreducible simply connected symmetric spaces of compact type and rank two. Also, we show a fact for the sections of equifocal submanifolds with non-flat section in other irreducible simply connected symmetric spaces of comp…
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.
In this paper, I prove a splitting theorem for equifocal submanifolds with non-flat section in a simply connected symmetric space of compact type. Also, by using the splitting theorem, I prove that the sections of equifocal submanifolds with non-flat section in an irreducible simply connected symmetric space of compact…
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.
It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.
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. Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
Taubes proved that all compact oriented four-manifolds admit non-flat instantons. We show that there exists a non-compact oriented four-manifold having no non-flat instanton.
Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
problem Integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
method Constructs a bihamiltonian integrable hierarchy of hydrodynamic type.
result Integrable hierarchy possesses Virasoro symmetries and a tau structure.
New Einstein metric found on non-standard solvmanifold.
problem Finding indefinite invariant Einstein metrics on solvmanifolds.
method Described an example of a non-standard solvmanifold with an indefinite invariant Einstein metric.
result Found a new type of Einstein metric on a non-standard solvmanifold.
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…
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.
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.
2-jets determine symmetries of parabolic geometries.
problem Determining symmetries of parabolic geometries.
method 2-jet determinacy for the symmetry algebra of parabolic geometries.
result Symmetry algebra is 1-jet determined at non-flat points and for various parabolic geometries.
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.
Study finds formulas for special curves in complex spaces.
problem Understanding special curves in complex spaces.
method Obtained explicit formulas for Killing magnetic curves.
result Explicit formulas for Killing magnetic curves in non-flat Lorentzian-Heisenberg spaces.
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.
The paper studies singularities in discrete indefinite affine minimal surfaces.
problem Characterizing singularities in discrete indefinite affine minimal surfaces.
method Discretizing smooth curves and applying discrete Lelieuvre's formulas to study the resulting surfaces.
result The definition of singular edges and vertices in discrete asymptotic nets mirrors properties of smooth surfaces.
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.
We study the basic geometric properties of an indefinite locally conformal Kaehler manifold.
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.
The paper extends and generalizes nullity distributions on real hypersurfaces in complex space forms.
problem Analyzing nullity distributions on real hypersurfaces in non-flat complex space forms.
method Extending and generalizing the concept of nullity distributions to three-dimensional cases and introducing new distributions.
result Results on real hypersurfaces in non-flat complex space forms, including new nullity distributions.
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…
Study on real hypersurfaces with special solitons in complex space forms.
problem Characterizing real hypersurfaces with ∗-Ricci solitons in non-flat complex space forms. method Analyzing real hypersurfaces with ∗-Ricci solitons where the potential vector field is in the principal curvature space and holomorphic distribution. result Characterized real hypersurfaces with ∗-Ricci solitons in non-flat complex space forms. We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
The study explores indefinite nilsolitons and Einstein solvmanifolds, revealing new geometric properties.
problem Understanding indefinite nilsolitons and their relation to Einstein solvmanifolds.
method Analyzing algebraic properties and geometric structures of nilpotent Lie algebras.
result Established a correspondence between indefinite Einstein solvmanifolds and nilsolitons, differing from the Riemannian case.
Enhances KLR for indefinite kernels with L1-norm regularization.
problem Classifying with indefinite kernels captures more domain-specific information.
method Introduces L1-norm regularization to induce sparsity and a proximal linearized algorithm. result Superior performance in accuracy and sparsity on multiple datasets.
Study on non-flat two-plectic geometry of six-sphere and its Hamiltonian dynamics.
problem Non-flat two-plectic geometry of six-sphere and Hamiltonian dynamics.
method Explicitly proving non-flatness and showing infinitesimal automorphisms via g2. result Explicit solutions of Hamilton-de Donder-Weyl equations with one- and two-dimensional sources.
Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.
The paper examines conditions of parallelism for a special tensor on 3D hypersurfaces.
problem Analyzing the conditions of parallelism for a specific tensor on 3D hypersurfaces.
method Extending existing results on real hypersurfaces with different types of ∗-Ricci tensor in complex space forms. result New findings on the ξ-parallelism of ∗-Ricci tensor on real 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.
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 …
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.
Paper tackles indefinite kernels in logistic regression.
problem Building logistic regression with indefinite kernels.
method Introduces IKLR model in RKKS, uses concave-inexact-convex procedure (CCICP) to solve non-convex optimization.
result Proposed method works effectively under deterministic and stochastic settings.
Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are know…