The paper establishes new Casorati inequalities for various Riemannian maps and submersions.
problem Developing new inequalities for Riemannian maps and submersions.
method Using general forms of Casorati inequalities, the paper derives inequalities for specific Riemannian spaces.
result The paper provides new Casorati inequalities for Riemannian maps and submersions.
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…
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.
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…
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.
Paper establishes new inequality for Riemannian maps and applies it to various space forms.
problem Developing a new inequality for Riemannian maps and its applications.
method Proposed and utilized a general Chen's first inequality for Riemannian maps and applied it to various space forms.
result Validated the new inequality and compared results with existing approaches.
The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.
problem Analyzing harmonic forms on almost Hermitian manifolds and complex surfaces.
method Using techniques from Bott-Chern and Aeppli numbers, the study generalizes harmonic forms from complex and symplectic manifolds to almost Hermitian manifolds.
result Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold.
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 …
The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.
problem Understanding cohomologies and harmonic forms on almost complex manifolds.
method Introducing new cohomologies (Bott-Chern and Aeppli) and studying associated harmonic forms.
result Bott-Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an invariant.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
Local holomorphic maps preserving (p,p) forms are shown to be isometries.
problem Preserving (p,p) forms under holomorphic maps between Kähler manifolds.
method Analyzing local holomorphic maps between Kähler manifolds, proving isometries up to scalars.
result Holomorphic maps preserving (p,p) forms are isometries under certain conditions.
New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic 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…
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.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
Study ruled real hypersurfaces in nonflat complex space forms with constant norm shape operators.
problem Proving nonexistence and characterizing ruled hypersurfaces in nonflat complex space forms.
method Analyzing shape operators with constant squared norm and using biharmonic properties.
result Biharmonic ruled real hypersurfaces are minimal, leading to their classification.
It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic spac…
Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.
problem Understanding real hypersurfaces in complex space forms and their properties.
method Investigating real hypersurfaces that achieve equality in a specific inequality involving a contact invariant.
result Characterized real hypersurfaces in complex space forms achieving the equality in the inequality.
Study on PMC surfaces in complex space forms, linking biconservative and totally real properties.
problem Characterizing PMC surfaces in complex space forms and their properties.
method Analyzing interactions between PMC, totally real, and biconservative properties; proving rigidity and reduction codimension results.
result PMC surfaces in non-flat complex space forms are biconservative if and only if totally real.
Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
Classifies toric dually flat manifolds into complex space forms.
problem Classifying 1D toric dually flat manifolds.
method Using complex space forms and exponential families.
result Toric dually flat manifolds are complex space forms.
The paper derives inequalities for Riemannian submersions and applies them to specific space forms.
problem Deriving inequalities for Riemannian submersions.
method Introducing and deriving inequalities for vertical, horizontal, and mixed distributions of Riemannian submersions.
result Established relationships between intrinsic and extrinsic invariants of Riemannian submersions.
The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.
problem Characterizing spheres in complex and Sasakian space forms.
method Establishing optimal integral inequalities involving Ricci curvature and second fundamental form norms.
result New characterizations of Whitney and contact Whitney spheres in complex and Sasakian space forms.
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. 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.
problem Analyzing extremal Kaehler metrics induced by complex space forms in finite and infinite dimensions.
method Proving properties of extremal Kaehler metrics under specific conditions in both finite and infinite dimensional settings.
result Extremal Kaehler metrics induced by infinite dimensional elliptic complex space forms have constant non-positive holomorphic sectional curvature under stability conditions.
The paper derives Chen-Ricci inequalities for Riemannian submersions and maps.
problem Chen-Ricci inequalities for Riemannian submersions and maps.
method General forms of Chen-Ricci inequalities for Riemannian submersions and maps are derived, involving curvatures of subspaces.
result New, easy, and elegant techniques for Chen-Ricci inequalities are established.
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…
Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.
problem Characterizing Kähler-Ricci solitons that can be immersed into complex space forms.
method Proving that a Kähler-Ricci soliton's metric is Einstein if it can be immersed into a complex space form.
result The Kähler-Ricci soliton's metric is an Einstein metric if it can be immersed into a complex space form.
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
Paper classifies special slant surfaces with varying curvature.
problem Classifying surfaces with non-constant mean curvature.
method Analyzing special slant surfaces in complex space forms.
result Complete classification of surfaces with non-constant mean curvature.
A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…
This is a complete classification of the complex forms of quaternionic symmetric spaces
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous 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.
problem Matching symplectic forms on circle pattern space.
method Pullback of symplectic forms to circle pattern space.
result Symplectic forms on circle pattern space coincide.
Simple proof for special surface classification.
problem Classifying surfaces with specific curvature properties.
method Elementary proof for parallel mean curvature surfaces.
result A lemma is proven for surfaces in complex space forms.
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…
We study f-biharmonic and bi-f-harmonic submanifolds in both generalized complex and Sasakian space forms. We prove necessary and sufficient condition for f-biharmonicity and bi-f-harmonicity in the general case and many particular cases. Some non-existence results are also obtained.
The paper sets new bounds for Ricci-curvature in submersions and maps.
problem Establishing bounds for Ricci-curvature in submersions and maps.
method Developed upper and lower bounds for Ricci-curvature in submersions and maps, providing geometric characterizations.
result New bounds for Ricci-curvature in submersions and maps have been derived.
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…
In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product…
The paper explores spaces of Kähler and symplectic forms on 4-manifolds.
problem Investigating the properties of Kähler and symplectic forms on 4-manifolds.
method Analyzing the uniqueness, connectedness, and openness of spaces of Kähler forms and introducing holomorphically tamed symplectic forms.
result Formulated a parallel question for holomorphically tamed symplectic forms and related it to Kähler-type symplectic forms.
The paper proves existence and uniqueness of slant immersions in complex space forms.
problem Existence and uniqueness of slant immersions in complex space forms.
method Established existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
result Existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
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.