arXiv research

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 give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functional…

In this paper we provide a method to study critical points of strongly indefinite functionals on vector bundles. We focus mainly on energy functionals coupled with a fermionic part, that is with a Dirac-type operator. We consider the cases of the perturbed Dirac-Geodesics problem and the Yang-Mills-Dirac type equation …

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 …

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

Introduces a variational framework for indefinite Lagrangians with specific symmetries.

We call a Morse function $f$ on a closed manifold $k$-constrained if neither $f$ nor $-f$ has critical points of indefinite Morse index $< k$. In this paper we study bordism groups of $k$-constrained Morse functions, and thus interpolate between the case $k = 1$ of bordism groups of Morse functions (computed by Ikegami…

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…

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…

Study of solutions for a spinorial Dirac equation with critical exponent.

New scalable methods for learning with indefinite kernels.

Study on lightlike geometry in indefinite Sasakian statistical manifolds.

In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to g…

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.

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more compli…

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…

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

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 prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we c…

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

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.

Study on special null submanifolds in indefinite Sasakian manifolds.

Formula derived for discrete improper affine spheres.

New Einstein metric found on non-standard solvmanifold.

In kernel methods, the kernels are often required to be positive definite, which restricts the use of many indefinite kernels. To consider those non-positive definite kernels, in this paper, we aim to build an indefinite kernel learning framework for kernel logistic regression. The proposed indefinite kernel logistic r…

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.

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.

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

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of qu…

In this paper we build the structure equations and the integrable systems for a discrete centroaffine indefinite surface in $\R^3$. At the same time, some centroaffine invariants are obtained according to the structure equations. Using these centroaffine invariants, we study the Laplacian operator and the convexity of …

The study explores indefinite nilsolitons and Einstein solvmanifolds, revealing new geometric properties.

Enhances KLR for indefinite kernels with $L_1$-norm regularization.

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…

Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.

Study on null submanifolds in indefinite complex contact geometry.

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.

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…

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

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…

New method reduces variance and bias in approximating indefinite kernels.

Study on quasilinear elliptic equation on graphs with indefinite weights.

Study on lightlike submanifolds in statistical manifold geometry.