arXiv research

Study on stochastic flows on 7-dimensional spheres.

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among …

We find the characterization of maximum dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds of a Sasakian space form and then classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere $\mathbb{S}^7$ we explicitly find all 3-dimensional proper-biharmonic integral $C$…

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric on the complement of an m-dimensional sphere in an n-dimensional sphere for some…

We deal with seven dimensional compact Riemannian manifolds of positive curvature which admit a cohomogeneity one action by a compact Lie group G. We prove that the manifold is diffeomorphic to a sphere if the dimension of the semisimple part of G is bigger than 6.

We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kähler manifold is endowed with an involutiv…

The volume of a k-dimensional foliation $\mathcal{F}$ in a Riemannian manifold $M^{n}$ is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on…

This paper contains all computations supporting the classification of 7-dimensional Einstein nilradicals given in the article "Classification of 7-dimensional Einstein nilradicals" (arXiv). Each algebra is analyzed in detail here.

In this paper we give the list of all 7-dimensional nilpotent real Lie algebras that admit a contact structure. Based on this list, we describe all 7-dimensional nilmanifolds that admit an invariant contact structure.

Proof of theorem for 7D manifolds with boundary using Witten deformation.

In this paper we give a generalisation of the Radius Rigidity theorem of F.Wilhelm. This is done by showing that if a Riemannian submersion of $S^{15}$ with 7-dimensional fibres has at least one fibre which is a great sphere then all the fibres are so. Some weaker than known conditions which force the existence of such…

We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not adm…

The paper proves rigidity for mapping class group actions on metrics of positive scalar curvature.

Study finite group actions on exotic aspherical space forms.

Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group SO_3 with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO_3's, one for each sphe…

In [5], D. Fetcu and C. Oniciuc presented the classification result for biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in $7$-dimensional Sasakian space forms. However, it is incomplete. In this paper, all such submanifolds are explicitly determined.

M. Kontsevich constructed universal characteristic classes of smooth bundles with fiber a framed odd-dimensional integral homology sphere. In dimension 3, they are known to give a universal finite type invariants of homology 3-spheres. However, they have not been well understood for higher fiber dimensions. The purpose…

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $\varphi$ such that the induced metric $g_{\varphi}$ is Einstein, unless $g_{\varphi}$ is flat.…

In this paper we show that a substantial Riemannian submersion of S(15) with 7- dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of of the Diameter Rigidity theorem for the Cayley plane which is considerably stronger than the very recent Radius Rigidity Theo…

Classifies nilpotent Lie groups with specific $G_2$-structures.

We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and …

Study on $G_2^*$ structures and almost para-contact structures in 7D.

We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure tha…

Study solutions and singularities of G2-structures flows on specific manifolds.

It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant of an amphichiral integral homology 3-sphere M vanishes. In this paper, we give a new direct proof of this vanishing property. For such an M, we construct a manifold pair (Y,Q) of dimensions 6 and 3 equipped with so…

The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.

We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty,T)$, with $0<T<+\infty$. However, for the modified Laplacian coflow, we prov…

The aim of this paper is to classify Ricci soliton metrics on $7$-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in [Transformation Groups, Volume 17, Number 3 (2012), 639--656]. To this end, we use the classification of $7$-dimensional real nilpotent Lie algebras given by Ming-Pe…

Study of singular curves in a specific type of hyperbolic distribution.

Classifies 7D manifolds with specific geometric properties.

Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.

It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along wit…

Study of special geometric structures on Lie groups.

We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup …

An infinite family of distinct harmonic morphisms with minimal circle fibers from the 7-dimensional homogeneous Allof-Wallach spaces of positive curvature onto the 6-dimensional flag manifolds is given.

The paper classifies foliations formed by generic coadjoint orbits of specific Lie groups.

We construct a new 7-dimensional manifold with positive sectional curvature which is 2-connected with π_3=\Z_2 and admits an isometric group action with one dimensional quotient.

We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.

Study heterotic G2-system on 2-step nilmanifolds with torus bundles.

The paper characterizes Pfaffian embeddings from 2,3,5-manifolds to 7-dimensional isotropic spaces.

Neural and numerical methods approximate G2-structures on Calabi-Yau manifolds.

The abstract describes a foliation of orbits for a specific class of Lie groups.

We apply E. Cartan's method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds $M$ up to local CR equivalence in the case that the cubic form of $M$ satisfies a certain symmetry property with respect to the Levi form of $M$. The solution to the equivalence problem is given by a parallelism on a prin…

Any 7-dimensional cocalibrated G_2-manifold admits a unique connection $\nabla$ with skew symmetric torsion. We study these manifolds under the additional condition that the $\nabla$-Ricci tensor vanishes. In particular, we describe their geometry in case of a maximal number of $\nabla$-parallel vector fields.

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize …

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $\varphi$ induced by the Calabi-Yau $3$-orbifold basic geometry. We disti…

We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then sh…

We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature, which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of posit…