arXiv research

Every real 3-manifold can be turned into a real contact structure.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…

Spin-structures on real Bott manifolds with Kähler structure are characterized.

Classifies real tight contact structures on lens spaces and solid tori.

Survey on hyperplane arrangements and their topology.

The study characterizes real flag manifolds with invariant generalized almost complex structures.

Study real logarithms of semi-simple matrices, focusing on differential structure.

We show that the connected sum of two copies of real projective 3-space does not admit a real projective structure. This is the first known example of a connected 3-manifold without a real projective structure.

Condition found for spinc structures on a specific type of manifold.

In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures.…

We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional directi…

New real invariants for 3-manifolds and links.

The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.

Study real projective structures on a specific Coxeter orbifold.

Real algebraic structures help classify overtwisted contact 3-spheres.

Developed a real sutured Heegaard Floer theory.

The paper classifies complex Dirac structures on flag manifolds.

Study classifies real hypersurfaces in complex projective spaces based on Lie derivatives and structure Jacobi operator properties.

Establishes correspondence between Calabi-Yau and Landau-Ginzburg structures.

It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply connected manifold $M$ which is not a sphere $(\dim M \geq 4)$ does not admit a real p…

We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.

We show that the group of smooth homotopy $7$-spheres acts freely on the set of smooth manifold structures on a topological manifold $M$ which is homotopy equivalent to the real projective $7$-space. We classify, up to diffeomorphism, all closed manifolds homeomorphic to the real projective $7$-space. We also show that…

First we introduce the notion of parallel structure Jacobi operator for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . Next we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with parallel structure Jacobi operator.

In this paper the result of real hypersurfaces in non-flat complex space forms, whose structure vector field $ξ$ belongs to the $κ$-nullity distribution is extended in case of three dimensional real hypersurfaces in non-flat complex space forms. Furthermore, generalization of notion ($κ$,$μ$)-nullity distribution defin…

New proof shows certain manifolds cannot have real projective structure.

Classifies complex Dirac structures with invariants and local structure.

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure…

In this paper we study the degeneration of convex real projective structures on bordered surfaces.

We give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan's $1$-minimal models of certain differential graded algebras associated with real variations of Hodge structures.

In this paper we prove some classification theorems of real hypersur- faces in Mn(c) satisfying certain conditions on the covariant derivative of the structure Jacobi operator. We also prove the non-existence of real hypersurfaces with Codazzi type structure Jacobi operator in Mn(c).

Study automorphisms and real structures on a special super-Grassmannian.

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Froben…

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…

A brief survey of real algebraic structures on topological spaces is given. This article is written for the Gokova Gemetry/Topology Conference proceedings.

Proof that convex structures on manifolds are open and closed.

Real Seiberg-Witten and monopole Floer homologies are equivalent for certain 3-manifolds.

Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.

Origami structures are enumerated and shown to be quantum modular.

Study examines tangential real hypersurfaces on Hermite-like manifolds.

We construct a smooth Lie group structure on the group of real analytic diffeomorphisms of a compact analytic manifold with corners. This generalises the known analogous results in the situation where the real analytic manifold has no corners. Additionally our approach uses a different construction.

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show th…

Geometric compactification for complex structures on Lie groups.

We introduce real structures on $L$-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$ is a real form of a connected semisimple complex affine algebraic group $G^{\mathbb{C}}$, constit…

We classify real hypersurfaces in CP^2and CH^2 equipped with pseudo-parallel structure Jacobi operator.

We investigate the $\mathrm{Pin}^-(2)$-monopole invariants of symplectic $4$-manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic $4$-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic $4$-manifolds. Further…

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…