arXiv research

Classifies actions on complex space forms with Lagrangian orbits.

New interpretation of complex hyperbolic form as Weil-Petersson form.

It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.

New curvature positivity helps classify spherical spaces and complex projective spaces.

New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.

Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.

Study on PMC surfaces in complex space forms, linking biconservative and totally real properties.

Study examines Kähler immersions of ALE Kähler metrics into complex space forms.

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denot…

The paper establishes new Casorati inequalities for various Riemannian maps and submersions.

Classifies toric dually flat manifolds into complex space forms.

We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature estimates. We consider curves, complex and Lagrangian surfaces and hypersurfaces for the…

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.

Computes tube formulas for valuations in complex space forms.

The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.

In this paper, we study a slant submanifold of a complex space form. We also obtain an integral formula of Simons' type for a Kaehlerian slant submanifold in a complex space form and apply it to prove our main result.

The paper studies extremal Kaehler metrics induced by complex space forms, proving properties in both finite and infinite dimensions.

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter…

Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.

Study Hamiltonian stationary Lagrangian surfaces in complex space forms.

Paper classifies special slant surfaces with varying curvature.

The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.

This is a complete classification of the complex forms of quaternionic symmetric spaces

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …

Symplectic forms match on circle pattern space.

Simple proof for special surface classification.

The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequal…

In this paper we first give a Bonnet theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two…

The paper explores spaces of Kähler and symplectic forms on 4-manifolds.

The paper proves existence and uniqueness of slant immersions in complex space forms.

This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in case…

We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.

The study finds minimal surfaces in complex space forms are often totally geodesic.

Lagrangian submanifolds of a Kaehler manifold are called Hamiltonian-stationary (or $H$-stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In [B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-sp…

We use the Grauert--Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski's sheaf of functions on the inertia space.

We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their…

Based on a well-known fact that there are no Einstein hypersurfaces in a non-flat complex space form, in this article we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hyersurface of a non-flat complex space form. For the real hypersurface with quasi-Einstein metric of …

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(φ, ξ, η, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_ξ=R(\cdot,ξ)ξ$ is $ξ$-parallel. In particular, we prove that the condition $\nabla_ξ R_ξ=0$ characterizes the …

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a…

Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex …

This paper proves a conjecture about Kähler manifolds and complex space forms.

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…

Paper establishes new inequality for Riemannian maps and applies it to various space forms.

It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.

Local holomorphic maps preserving (p,p) forms are shown to be isometries.