arXiv subjects

Zuzanna Szancer

Publications and source records attributed to Zuzanna Szancer.

7 recordsLinked to original sources

arXiv2026

Contact variation of almost contact pseudo-metric structures

For an almost contact pseudo-metric manifold we study a variation of the metric in the spirit of Meli-Ngakeu-Olea. Such variation is defined using a vector field on the manifold and the procedure shifts the signature of the metric by increasing the positive index by two. First we derive conditions on the vector field for the obtained manifold to remain almost contact pseudo-metric. Then we study the cases of almost pseudo co-Kähler manifolds. In particular we obtain geometric characterizations of variations for which the fundamental form remains closed and establish a local decomposition into a product. We further investigate the variation in the setting of statistical structures.

math.DG↗

arXiv2021

Affine hypersurfaces of arbitrary signature with an almost symplectic form

In this paper we study affine hypersurfaces with non-degenerate second fundamental form of arbitrary signature additionally equipped with an almost symplectic structure $ω$. We prove that if $R^pω=0$ or $\nabla^pω=0$ for some positive integer $p$ then the rank of the shape operator is at most one. The results provide complete classification of affine hypersurfaces with higher order parallel almost symplectic forms and are generalization of recently obtained results for Lorentzian affine hypersurfaces.

math.DG↗

arXiv2019

On $\widetilde{J}$-tangent affine hyperspheres

In this paper we study $\widetilde{J}$-tangent affine hyperspheres, where $\widetilde{J}$ is the canonical para-complex structure on $\mathbb{R}^{2n+2}$. The main purpose of this paper is to give a classification of $\widetilde{J}$-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution $\mathcal{D}$. In particular, we classify all such hyperspheres in the $3$-dimensional case. We also show that there is a direct relation between $\widetilde{J}$-tangent affine hyperspheres and Calabi products. As an application we obtain certain classification results. In particular, we show that, with one exception, all odd dimensional proper flat affine hyperspheres are, after a suitable affine transformation, $\widetilde{J}$-tangent. Some examples of $\widetilde{J}$-tangent affine hyperspheres are also given.

math.DG↗

arXiv2018

On $4$-dimensional Lorentzian affine hypersurfaces with an almost symplectic form

In this paper we study $4$-dimensional affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure $ω$. We prove that the rank of the shape operator is at most one if $R^k\cdot ω=0$ or $\nabla^kω=0$ for some positive integer $k$. This result is the final step in a classification of Lorentzian affine hypersurfaces with higher order parallel almost symplectic forms.

math.DG↗

arXiv2018

Parallel Almost Paracontact Structures on Affine Hypersurfaces

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent transversal vector field. Such vector field induces in a natural way an almost paracontact structure ${(φ,ξ,η)}$ on $M$ as well as the affine connection $\nabla$. In this paper we give the classification of hypersurfaces with the property that $φ$ or $η$ is parallel relative to the connection $\nabla$. Moreover, we show that if $\nablaφ=0$ (respectively $\nablaη=0$) then around each point of $M$ there exists a parallel almost paracontact structure. Results we illustrate with some examples.

math.DG↗

arXiv2018

On $3$-dimensional $\widetilde{J}$-tangent centro-affine hypersurfaces and $\widetilde{J}$-tangent affine hyperspheres with some null-directions

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of this paper is to give a full local classification of the above mentioned hypersurfaces. In particular, we prove that every nondegenerate centro-affine hypersurface of dimension $3$ with a $\widetilde{J}$-tangent centro-affine vector field which has two null-directions $\mathcal{D}^+$ and $\mathcal{D}^-$ must be both an affine hypersphere and a hyperquadric. Some examples of these hypersurfaces are also given.

math.DG↗

arXiv2017

$\widetilde{J}$-tangent affine hypersurfaces with an induced almost paracontact structure

We study real affine hypersurfaces $f\colon M\rightarrow \mathbb{R}^{2n+2}$ with an almost paracontact structure $(φ,ξ,η)$ induced by a $\widetilde{J}$-tangent transversal vector filed, where $\widetilde{J}$ is the canonical paracomplex structure on $\mathbb{R}^{2n+2}$. We give a classification of hypersurfaces for which an induced almost paracontact structure is metric relative to the second fundamental form. Some other properties of such hypersurfaces are also studied.

math.DG↗