arXiv research

In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in $C_k^m$. We introduce the notion of indefinite special Lagrangian submanifolds in $C_k^m$ and generalize the well-known work of Harvey-Lawson to the indefinite case.

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…

We present a compared analysis of some properties of indefinite almost $\mathcal{S}$-manifolds and indefinite $\mathcal{S}$-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 $\…

We show that an indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form. We further study whether two compact Fubini-Study spaces are relatives or not.

Study on special null submanifolds in indefinite Sasakian manifolds.

Twistor correspondences for R-invariant indefinite self-dual conformal structures on R^4 are established explicitly. These correspondences are written down by using a natural integral transform from functions on a two dimensional cylinder to functions on the flat Lorentz space R^{1,2} which is related to the wave equat…

The paper studies timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.

This paper studies ruled real hypersurfaces in indefinite complex projective space.

The study proves that pseudo-umbilical spacelike submanifolds are totally geodesic and totally umbilical.

Study new Hopf real hypersurfaces in indefinite complex projective space.

New types of null hypersurfaces found in Sasakian space-forms.

Following the approach to pseudo-Riemannian symmetric spaces developed in math.DG/0408249 we exhibit examples of indefinite hyper-Kaehler symmetric spaces with non-abelian holonomy. Moreover, we classify indecomposable hyper-Kaehler symmetric spaces whose metric has signature (4,4n). Such spaces exist if and only if n=…

New scalable methods for learning with indefinite kernels.

The study extends Jacobi-orthogonality to indefinite scalar product spaces.

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…

Study of regularized least squares in RKKS with indefinite kernels.

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 …

Paper tackles indefinite kernels in logistic regression.

Study of spheres and circles on a manifold with a specific metric structure.

Study on lightlike geometry in indefinite Sasakian statistical manifolds.

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…

Study on null hypersurfaces in complex contact manifolds.

Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.

Introduces new lightlike submanifolds in indefinite Kaehler manifolds.

Study lightlike submanifolds in indefinite statistical manifolds, finding conditions and curvature expressions.

Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.

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 …

Study Lie algebras of indefinite spaces with finite volume.

Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.

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.

Paper develops a new method for distribution regression with indefinite kernels.

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we su…

Formula derived for discrete improper affine spheres.

We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H…

New Einstein metric found on 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…

We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitr…

Improved indefinite Kasparov modules for non-elliptic operators.

We determine all indecomposable pseudo-Riemannian symmetric spaces of signature (4,3) whose holonomy is contained in $G_{2(2)}\subset SO(4,3)$.

In a noncompact harmonic manifold $M$ we establish finite dimensionality of the eigenspaces $V_λ$ generated by radial eigenfunctions of the form $\cosh r + c$. As a consequence, for such harmonic manifolds, we give an isometric imbedding of $M$ into $(V_λ,B)$, where $B$ is a nondegenerate symmetric bilinear indefinite …

Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.

Study on SASI-lightlike submanifolds in indefinite Kaehler manifolds.

The paper studies singularities in discrete indefinite affine minimal surfaces.

Introduces a variational framework for indefinite Lagrangians with specific symmetries.

We study the basic geometric properties of an indefinite locally conformal Kaehler manifold.

This thesis extends noncommutative geometry to semi-Riemannian manifolds and applies it to gauge theories.

New indefinite false theta functions match homological blocks for a specific 3-manifold.

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…