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Stefan Ivanov

Publications and source records attributed to Stefan Ivanov.

At least 19 recordsLinked to original sources
arXiv2026

Torsion and curvature of ACYT and AHKT 8-manifold

We observe that on a compact almost Hermitian 8 manifold with totally skew-symmetric Nijenhuis tensor parallel with respect to the unique almost hermitian connection with torsion three form if the three Ricci tensors of this connection vanish then the torsion 3-form is closed. Consequently, on a compact complex 8-manifold if the three Ricci tensors of the Strominger-Bismut connection vanish then its torsion is closed, i.e. the space is pluriclosed. We also deduce that a compact ACYT 8-manifold with paralle Nijenhuis tensor with respect to the torsion connection has closed torsion if and only if the trace of the exterior derivative of the torsion vanishes. It is shown that a compact ACYT 8-manifold wich Nijenhuis tensor is parallel with respect to the torsion connection is a Ricci flat $SU(4)$ instanton exactly when the torsion 3-form is parallel with respect to the torsion and to the Levi-Civita connections simultaneously. We consider an almost hyperhermitian 8-manifold with an $Sp(2)$ structure admitting linear connection preserving the $Sp(2)$ structure and having totally skew symmetric torsion and call it AHKT manifold. The exact conditions an almost hyperhermitian 8-manifold to be an AHKT are presented and it is shown that an AHKT 8-manifold is HKT exactly when the three Lee forms coincide

math.DG↗
arXiv2025

Parallel torsion and $G_2, Spin(7)$ instantons

Instanton properties of the characteristic connection $\nabla$ on an integrable $G_2$ manifold as well as instanton condition of the torsion connection $\nabla$ on a $Spin(7)$ manifold are investigated. It is shown that for an integrable $G_2$ manifold with $\nabla$-parallel Lee form the curvature of the characteristic connection is a $G_2$ instanton exactly when the torsion 3-form is $\nabla$-parallel. It is observed that on a compact $Spin(7)$ manifold with $\nabla$ closed torsion 3-form the torsion connection is a $Spin(7)$ instanton if and only if the torsion 3-form is parallel with respect to the torsion connection.

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arXiv2025

The Riemannian curvature identities of a $G_2$ connection with skew-symmetric torsion and generalized Ricci solitons

Curvature properties of the characteristic connection on an integrable $G_2$ manifold are investigated. We consider integrable $G_2$ manifold of constant type, i.e. the scalar product of the exterior derivative of the $G_2$ form with its Hodge dual is a constant. We show that on an integrable $G_2$ manifold of constant type with $G_2$-instanton characteristic curvature and vanishing Ricci tensor the torsion 3-form is harmonic. Consequently, we prove that the characteristic curvature is symmetric in exchange the first and the second pair and Ricci flat if and only if the three-form torsion is parallel with respect to the Levi-Civita and to the characteristic connection simultaneously and this is equivalent to the condition that the characteristic curvature satisfies the Riemannian first Bianchi identity. We find that the Hull connection is a $G_2$-instanton exactly when the torsion is closed. We observe that any compact integrable $G_2$ manifold with closed torsion is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the characteristic connection. In particular, this vector field is an infinitesimal automorphism of the $G_2$ structure and preserves the torsion three form.

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arXiv2025

The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons

It is shown that on compact $Spin(7)$--manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$--torsion connection $R\in S^2Λ^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$--torsion connections simultaneously. Precise conditions for a compact $Spin(7)$--manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$--torsion connection. It is shown that a compact $Spin(7)$--manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$--manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$--structure.

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arXiv2025

Almost Calabi-Yau with torsion 6-manifolds and the instanton condition

It is observed that on a compact almost complex Calabi-Yau with torsion (ACYT) 6-manifold with co-closed Lee form the curvature of the torsion connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the torsion connection. The same conclusion holds for any (non necessarily compact) balanced ACYT 6-manifold. In particular, on a CYT 6-manifold the Strominger-Bismut connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the Strominger-Bismut connection provided either the CYT 6-manifold is compact with co-closed Lee form or it is a balanced CYT 6-manifold.

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arXiv2024

The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=δT=0$ or the curvature of the torsion connection $R\in S^2Λ^2$ then the scalar curvature of a $\nabla$-Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity $R(X,Y,Z,V)=R(Z,Y,X,V)$ must be flat.

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arXiv2024

The Riemannian curvature identities on almost Calabi-Yau with torsion 6-manifold and generalized Ricci solitons

It is observed that on a compact almost complex Calabi-Yau with torsion 6-manifold the Nijenhuis tensor is parallel with respect to the torsion connection. If the torsion is closed then the space is a compact generalized gradient Ricci soliton. In this case, the torsion connection is Ricci-flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. On a compact almost complex Calabi-Yau with torsion 6-manifold it is shown that the curvature of the torsion connection is symmetric on exchange of the first and the second pairs and has vanishing Ricci tensor if and only if it satisfies the Riemannian first Bianchi identity.

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arXiv2024

Li-Yau sub-gradient estimates and Perelman-type entropy formulas for the heat equation in quaternionic contact geometry

We establish in the present paper two sub-gradient estimates for the quaternionic contact (qc) heat equation on a compact qc manifold of dimension $4n+3$, provided some positivity conditions are satisfied. These are qc versions of the prominent Li-Yau gradient estimate in Riemannian geometry. Another goal of this paper is to get two Perelman-type entropy formulas for the qc heat equation on a compact qc-Einstein manifold of dimension $4n+3$ with non-negative qc scalar curvature (e.g. compact $3$-Sasakian manifold), as well as an integral sub-gradient estimate for the positive solutions of the qc heat equation.

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arXiv2024

Conformal paraquaternionic contact curvature and the local flatness theorem

A tensor invariant is defined on a paraquaternionic contact manifold in terms of the curvature and torsion of the canonical paraquaternionic connection involving derivatives up to third order of the contact form. This tensor, called paraquaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry, the Chern-Moser tensor in CR geometry, the para contact curvature in para CR geometry and to the quaternionic contact conformal curvature in quaternionic contact geometry. It is shown that a paraquaternionic contact manifold is locally paraquaternionic contact conformal to the standard flat paraquaternionic contact structure on the paraquaternionic Heisenberg group, or equivalently, to the standard para 3-Sasakian structure on the paraquaternionic pseudo-sphere iff the paraquaternionic contact conformal curvature vanishes.

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arXiv2024

Geometry of paraquaternionic contact structures

We introduce the notion of paraquaternionic contact structures (pqc structures), which turns out to be a generalization of the para 3-Sasakian geometry. We derive a distinguished linear connection preserving the pqc structure. Its torsion tensor is expressed explicitly in terms of the structure tensors and the structure equations of a pqc manifold are presented. We define pqc-Einstein manifolds and show that para 3-Sasakian spaces are precisely pqc manifolds, which are pqc-Einstein. Furthermore, we introduce the paraquaternionic Heisenberg qroup and show that it is the flat model of the pqc geometry.

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arXiv2022

The Almost Schur Lemma in Quaternionic Contact Geometry

We establish quaternionic contact (qc) versions of the so called Almost Schur Lemma, which give estimations of the qc scalar curvature on a compact qc manifold to be a constant in terms of the norm of the $[-1]$-component and the norm of the trace-free part of the $[3]$-component of the horizontal qc Ricci tensor and the torsion endomorphism, under certain positivity conditions.

math.DG↗
arXiv2022

The CR Almost Schur Lemma and the positivity conditions

We establish a new version of the CR almost Schur Lemma which gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold to be a constant in terms of the norm of the traceless Webster Ricci tensor and the pseudohermitian torsion under a certain positivity condition. In the torsion-free case, i.e. for a compact Sasakian manifold, our positivity condition coincides with the known one and we obtain a better estimate

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arXiv2020

Para-Sasaki-like Riemannian manifolds and new Einstein metrics

We extract a new class of paracontact paracomplex Riemannian manifolds arising from certain cone construction, call it para-Sasaki-like Riemannian manifold and give explicit examples. We define a hyperbolic extension of a paraholomorphic paracomplex Riemannian manifold, which is a local product of two Riemannian spaces with equal dimensions, showing that it is a para-Sasaki-like Riemannian manifold. If the starting paraholomorphic paracomplex Riemannian manifold is complete Einstein with negative scalar curvature then its hyperbolic extension is a complete Einstein para-Sasaki-like Riemannian manifold with negative scalar curvature thus producing new examples of complete Einstein Riemannian manifold with negative scalar curvature.

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arXiv2019

Non-symmetric Riemannian gravity and Sasaki-Einstein 5-manifolds

We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condition is equivalent to the existence of a Sasaki-Einstein 5-manifold and vice versa, any Sasaki-Einstein 5-manifold generates a two parametric family of connections with skew torsion satisfying the Einstein metricity condition. Formulas for the curvature and the Ricci tensors of these connections are presented in terms of the Sasaki-Einstein SU(2) structures.

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arXiv2018

A Bonnet-Myers type theorem for quaternionic contact structures

We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.

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arXiv2017

On the Strominger system and holomorphic deformations

We show that the property of existence of solution to the Strominger system in dimension six is neither open nor closed under holomorphic deformations of the complex structure. These results are obtained both in the case of positive slope parameter as well as in the case of negative slope parameter in the anomaly cancellation equation.

math.DG↗
arXiv2016

The qc Yamabe problem on non-spherical quaternionic contact manifolds

It is shown that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Namely, it is proved that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature

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